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Factoring Polynomials

42 questions

All types42Multiple choice22Matching7Fill in the blank8Short answer5
Question 1Question →

Consider the algebraic expression: 3x2+6x−243x^2 + 6x - 24. When this expression is factored completely, which of the following represents the correct product?

Show answer & explanation

Answer: 3(x+4)(x−2)3(x + 4)(x - 2)

Answer

The completely factored form is 3(x+4)(x−2)3(x + 4)(x - 2).
To factor the expression completely, you must first factor out the greatest common factor (GCF) of 3 from all terms. This leaves 3(x2+2x−8)3(x^2 + 2x - 8). Next, you factor the quadratic trinomial inside the parentheses. You need two numbers that multiply to -8 and add to +2. Those numbers are +4 and -2. Therefore, combining the GCF and the binomials gives the completely factored form: 3(x+4)(x−2)3(x + 4)(x - 2).

Step-by-Step Solution

1
Identify and factor out the greatest common factor (GCF) from all three terms in the expression 3x2+6x−243x^2 + 6x - 24.
The GCF is 3. Factoring it out yields 3(x2+2x−8)3(x^2 + 2x - 8).
Factoring out the GCF simplifies the remaining trinomial, making it easier to factor and ensuring the final answer is factored completely.
2
Factor the remaining quadratic trinomial x2+2x−8x^2 + 2x - 8 by finding two integers that multiply to the constant term (-8) and add up to the middle coefficient (2).
The correct integers are 4 and -2 (since 4 * -2 = -8 and 4 + -2 = 2). This gives the binomial factors (x+4)(x−2)(x + 4)(x - 2).
This process decomposes the monic quadratic trinomial into its constituent binomial factors.
3
Combine the GCF and the binomial factors to write the final factored expression.
The final product is 3(x+4)(x−2)3(x + 4)(x - 2).
This represents the original expression as a product of its simplest prime factors.

Key Concept

Factoring Polynomials using GCF and Trinomial Decomposition
Question 2Question →

Determine the missing constants in the completely factored form of the given polynomial.

Fill in the blanks below

The polynomial 5x2−5x−605x^2 - 5x - 60 can be factored completely. When written in the form a(x−b)(x+c)a(x - b)(x + c), where bb and cc are positive integers, the value of aa is , the value of bb is , and the value of cc is .
Show answer & explanation

Answer

The constants are a = 5, b = 4, and c = 3.
The correct values are found by first factoring out the greatest common factor of 55, leaving the quadratic expression x2−x−12x^2 - x - 12. This quadratic expression factors into two binomials by finding two numbers that multiply to −12-12 and add to −1-1, which are −4-4 and 33. This yields the fully factored form 5(x−4)(x+3)5(x - 4)(x + 3). By matching this result to the required algebraic structure a(x−b)(x+c)a(x - b)(x + c), we can directly identify that a=5a = 5, b=4b = 4, and c=3c = 3.

Step-by-Step Solution

1
Find the greatest common factor (GCF) of the terms in the polynomial 5x2−5x−605x^2 - 5x - 60.
The GCF is 55. Factoring it out gives 5(x2−x−12)5(x^2 - x - 12).
Extracting the GCF simplifies the trinomial and makes it easier to factor into binomials.
2
Factor the remaining quadratic trinomial x2−x−12x^2 - x - 12.
We look for two integers that multiply to −12-12 and add to −1-1.
This is the standard procedure for factoring a monic quadratic trinomial of the form x2+bx+cx^2 + bx + c.
3
Identify the proper integers and write the binomial factors.
The integers are −4-4 and 33. The factored trinomial is (x−4)(x+3)(x - 4)(x + 3).
(−4)×3=−12(-4) \times 3 = -12 and −4+3=−1-4 + 3 = -1.
4
Combine the factors and match them to the target form a(x−b)(x+c)a(x - b)(x + c).
The completely factored expression is 5(x−4)(x+3)5(x - 4)(x + 3). Matching this to a(x−b)(x+c)a(x - b)(x + c) yields a=5a = 5, b=4b = 4, and c=3c = 3.
This establishes the precise values corresponding to the blanks in the question.

Key Concept

Factoring out a greatest common factor followed by factoring a quadratic trinomial.
Question 3Question →

The area of a rectangular solar panel is modeled by the algebraic expression w2−5w+6w^2 - 5w + 6. Which of the following represents the completely factored form of this expression?

Show answer & explanation

Answer: (w−2)(w−3)(w - 2)(w - 3)

Answer

The completely factored form of the expression is (w−2)(w−3)(w - 2)(w - 3).
To factor the quadratic expression w2−5w+6w^2 - 5w + 6, you must find two integers that multiply to the constant term (6) and add to the middle term's coefficient (-5). The integers -2 and -3 satisfy both conditions because (−2)×(−3)=6(-2) \times (-3) = 6 and (−2)+(−3)=−5(-2) + (-3) = -5. Placing these into binomials gives the correct factored form.

Step-by-Step Solution

1
Identify the type of expression
The expression is a quadratic trinomial in the form w2+bw+cw^2 + bw + c, where b=−5b = -5 and c=6c = 6.
Recognizing the structure helps determine the factoring method.
2
Find the factors of the constant term
The factors of 6 are (1, 6), (-1, -6), (2, 3), and (-2, -3).
The binomial factors must contain two numbers that multiply to the constant term.
3
Select the factor pair that adds to the middle coefficient
The sum of -2 and -3 is -5.
The two numbers must add up to the coefficient of the middle term.
4
Write the final factored expression
(w−2)(w−3)(w - 2)(w - 3)
Substituting the correct numbers into the binomial format (w+p)(w+q)(w + p)(w + q).

Key Concept

Factoring Monic Quadratic Trinomials
Question 4Question →

Match each polynomial expression on the left with its completely factored equivalent on the right.

Click a left item, then click its matching right item

Items

x2−3x−10x^2 - 3x - 10
2x2−182x^2 - 18
3x2−6x3x^2 - 6x
x2−10x+25x^2 - 10x + 25

Matches

Show answer & explanation

Answer

The correct matches pair each polynomial with its completely factored form by applying techniques such as factoring out the greatest common factor (GCF), recognizing difference of squares, and factoring trinomials.
Each polynomial is completely factored by extracting any Greatest Common Factors and correctly applying fundamental algebraic patterns, including the difference of squares, perfect square trinomials, and standard quadratic trinomial factoring.

Step-by-Step Solution

1
Analyze the first polynomial: x2−3x−10x^2 - 3x - 10.
It factors to (x−5)(x+2)(x - 5)(x + 2).
This is a basic trinomial where we need factors of -10 that add up to -3.
2
Analyze the second polynomial: 2x2−182x^2 - 18.
It factors to 2(x−3)(x+3)2(x - 3)(x + 3).
Factoring out the GCF of 2 leaves x2−9x^2 - 9, which is a difference of squares and factors further into (x−3)(x+3)(x - 3)(x + 3).
3
Analyze the third polynomial: 3x2−6x3x^2 - 6x.
It factors to 3x(x−2)3x(x - 2).
The two terms share a common factor of 3x3x. Extracting this GCF directly yields the factored form.
4
Analyze the fourth polynomial: x2−10x+25x^2 - 10x + 25.
It factors to (x−5)2(x - 5)^2.
This matches the perfect square trinomial pattern a2−2ab+b2=(a−b)2a^2 - 2ab + b^2 = (a - b)^2 with a=xa=x and b=5b=5.

Key Concept

Factoring Polynomials
Question 5Question →
Find the completely factored form of the given polynomial:
2x2+2x−402x^2 + 2x - 40

Assume all missing values in the blanks are positive integers.

Fill in the blanks below

The completely factored expression is (x + )(x - ).
Show answer & explanation

Answer

The missing values are 2, 5, and 4. The completely factored expression is 2(x + 5)(x - 4).
Factoring out the greatest common factor of 2 leaves x2+x−20x^2 + x - 20. This trinomial factors into (x+5)(x−4)(x + 5)(x - 4) because 5 and -4 multiply to give -20 and add to give 1. Thus, the complete factored form is 2(x+5)(x−4)2(x + 5)(x - 4), corresponding to the blank values 2, 5, and 4.

Step-by-Step Solution

1
Factor out the greatest common factor (GCF) from all the terms in the polynomial.
2(x2+x−20)2(x^2 + x - 20)
Identifying and extracting the GCF of 2 simplifies the remaining trinomial, making it easier to factor.
2
Factor the resulting quadratic trinomial x2+x−20x^2 + x - 20.
Find two numbers that multiply to -20 and add to 1 (the coefficient of x).
To factor a trinomial of the form x2+bx+cx^2 + bx + c, you must find two factors of cc that add up to bb.
3
Identify the two numbers that satisfy the conditions.
The numbers are 5 and -4.
5×(−4)=−205 \times (-4) = -20 and 5+(−4)=15 + (-4) = 1.
4
Write the complete factorization by combining the GCF and the factored binomials.
2(x+5)(x−4)2(x + 5)(x - 4)
This represents the original polynomial completely factored into its simplest terms.

Key Concept

Factoring Polynomials with a Greatest Common Factor
Question 6Question →

The total revenue from selling a batch of electronic components is modeled by the polynomial 15x2−14x−815x^2 - 14x - 8. If the number of units sold is given by the expression (5x+2)(5x + 2), what binomial represents the price per unit?

Show answer & explanation

Answer: 3x-4; (3x-4); 3x - 4; (3x - 4)

Answer

3x - 4
The correct answer is derived by factoring the revenue polynomial 15x2−14x−815x^2 - 14x - 8. Because revenue equals the number of units multiplied by the price per unit, the missing price expression is the binomial that, when multiplied by (5x+2)(5x + 2), yields the given polynomial. By factoring the expression, we find that 15x2−14x−8=(5x+2)(3x−4)15x^2 - 14x - 8 = (5x + 2)(3x - 4), confirming (3x−4)(3x - 4) is the correct binomial.

Step-by-Step Solution

1
Set up the algebraic relationship based on the context.
Total Revenue = (Number of Units) × (Price per Unit), which translates to 15x2−14x−8=(5x+2)×(Price per Unit)15x^2 - 14x - 8 = (5x + 2) \times (\text{Price per Unit}).
This establishes that the price per unit is the missing factor of the given polynomial.
2
Determine the factoring method for the trinomial 15x2−14x−815x^2 - 14x - 8.
Using the ac-method, multiply the leading coefficient (15) by the constant term (-8) to get −120-120.
This is the standard approach for factoring a trinomial where the leading coefficient is greater than 1.
3
Find two integers that multiply to −120-120 and add to the middle coefficient, −14-14.
The required integers are −20-20 and 66, because −20×6=−120-20 \times 6 = -120 and −20+6=−14-20 + 6 = -14.
These integers allow us to correctly split the middle term for factoring by grouping.
4
Rewrite the trinomial and factor by grouping.
15x2−20x+6x−8=5x(3x−4)+2(3x−4)=(5x+2)(3x−4)15x^2 - 20x + 6x - 8 = 5x(3x - 4) + 2(3x - 4) = (5x + 2)(3x - 4).
Grouping extracts the greatest common factor from pairs of terms, revealing the two binomial factors.
5
Identify the missing factor.
Since one factor is given as (5x+2)(5x + 2), the other factor representing the price is (3x−4)(3x - 4).
This completes the solution, satisfying the initial equation.

Key Concept

Factoring Non-Monic Trinomials
Question 7Question →

Factor the trinomial x2+2x−15x^2 + 2x - 15. Which of the following choices shows the correct factorization?

Show answer & explanation

Answer: (x+5)(x−3)(x + 5)(x - 3)

Answer

The correct factorization is (x+5)(x−3)(x + 5)(x - 3).
To factor the quadratic expression x2+2x−15x^2 + 2x - 15, we need to find two numbers that multiply to −15-15 and add up to 22. The numbers 55 and −3-3 meet these criteria (5×−3=−155 \times -3 = -15 and 5+−3=25 + -3 = 2). Therefore, the completely factored form is (x+5)(x−3)(x + 5)(x - 3).

Step-by-Step Solution

1
Identify the target sum and product for the factors.
The target sum is the coefficient of the middle term (22), and the target product is the constant term (−15-15).
To factor a trinomial x2+bx+cx^2 + bx + c into (x+p)(x+q)(x + p)(x + q), we need to find two integers pp and qq that multiply to cc and add up to bb.
2
Find the two integers that satisfy both conditions.
The factor pairs of −15-15 are (−1,15)(-1, 15), (1,−15)(1, -15), (−3,5)(-3, 5), and (3,−5)(3, -5). The pair that adds to 22 is −3-3 and 55.
Checking the sums, only 5+(−3)=25 + (-3) = 2 provides the correct middle coefficient.
3
Write the factored form using the selected integers.
The factored form is (x+5)(x−3)(x + 5)(x - 3).
Substituting 55 and −3-3 into the binomial framework (x+p)(x+q)(x + p)(x + q) completes the factorization.

Key Concept

Factoring Monic Quadratic Trinomials
Question 8Question →

Simplify the given algebraic expression by factoring it completely, then determine the missing positive integer values.

Fill in the blanks below

The completely factored form of the expression 2y2−10y−282y^2 - 10y - 28 can be written in the format 2(y−a)(y+b)2(y - a)(y + b), where aa and bb are positive integers.

The value of aa is and the value of bb is .
Show answer & explanation

Answer

The value of a is 7 and the value of b is 2.
Factoring out the GCF of 2 yields 2(y2−5y−14)2(y^2 - 5y - 14). The trinomial y2−5y−14y^2 - 5y - 14 factors into (y−7)(y+2)(y - 7)(y + 2). Matching this with the given structure 2(y−a)(y+b)2(y - a)(y + b) shows that the positive integers are a=7a = 7 and b=2b = 2.

Step-by-Step Solution

1
Factor out the greatest common factor (GCF) from all terms in the polynomial.
2(y2−5y−14)2(y^2 - 5y - 14)
Identifying and factoring out the GCF of 2 simplifies the polynomial, revealing a trinomial with a leading coefficient of 1, which is easier to factor.
2
Factor the resulting quadratic trinomial y2−5y−14y^2 - 5y - 14.
The binomial factors are (y−7)(y - 7) and (y+2)(y + 2).
To factor y2−5y−14y^2 - 5y - 14, we need to find two numbers that multiply to the constant term (-14) and add up to the middle term's coefficient (-5). The numbers -7 and 2 satisfy both conditions (-7 * 2 = -14 and -7 + 2 = -5).
3
Write the complete factorization and match it to the provided template 2(y−a)(y+b)2(y - a)(y + b).
The expression is 2(y−7)(y+2)2(y - 7)(y + 2). Matching this to the template gives a=7a = 7 and b=2b = 2.
Comparing the forms directly, the term subtracted from y is 7 (so a = 7) and the term added to y is 2 (so b = 2).

Key Concept

Factoring polynomials using the Greatest Common Factor (GCF) and factoring quadratic trinomials.
Question 9Question →

Match each non-monic quadratic polynomial to its completely factored form. Pay close attention to the signs in each binomial factor.

Click a left item, then click its matching right item

Items

2m2−11m+152m^2 - 11m + 15
2m2+m−152m^2 + m - 15
2m2+11m+152m^2 + 11m + 15
2m2−m−152m^2 - m - 15

Matches

Show answer & explanation

Answer

The correct pairings are: 2m2−11m+152m^2 - 11m + 15 with (2m−5)(m−3)(2m - 5)(m - 3); 2m2+m−152m^2 + m - 15 with (2m−5)(m+3)(2m - 5)(m + 3); 2m2+11m+152m^2 + 11m + 15 with (2m+5)(m+3)(2m + 5)(m + 3); and 2m2−m−152m^2 - m - 15 with (2m+5)(m−3)(2m + 5)(m - 3).
Each polynomial requires finding two numbers that multiply to the product of the leading coefficient and the constant term, and add to the middle coefficient. The correct pairings map the expanded polynomials directly to their unique, fully factored equivalents.

Step-by-Step Solution

1
Analyze the signs of the constants in each polynomial.
Polynomials with a positive constant term (+15) must have binomial factors with identical signs (both positive or both negative). Polynomials with a negative constant term (-15) must have binomial factors with opposite signs.
The product of the constant terms in the binomials must equal the constant term of the polynomial.
2
Factor 2m2−11m+152m^2 - 11m + 15.
Since the constant is positive and the middle term is negative, both factors must be negative. 2m2−6m−5m+15=2m(m−3)−5(m−3)=(2m−5)(m−3)2m^2 - 6m - 5m + 15 = 2m(m - 3) - 5(m - 3) = (2m - 5)(m - 3).
Applying factoring by grouping.
3
Factor 2m2+m−152m^2 + m - 15.
The constant is negative, so signs differ. The middle term is positive, so the larger product in FOIL must be positive. 2m2+6m−5m−15=2m(m+3)−5(m+3)=(2m−5)(m+3)2m^2 + 6m - 5m - 15 = 2m(m + 3) - 5(m + 3) = (2m - 5)(m + 3).
Applying factoring by grouping.
4
Factor 2m2+11m+152m^2 + 11m + 15.
Since both the constant and middle term are positive, both factors must be positive. 2m2+6m+5m+15=2m(m+3)+5(m+3)=(2m+5)(m+3)2m^2 + 6m + 5m + 15 = 2m(m + 3) + 5(m + 3) = (2m + 5)(m + 3).
Applying factoring by grouping.
5
Factor 2m2−m−152m^2 - m - 15.
The constant is negative, so signs differ. The middle term is negative, so the larger product in FOIL must be negative. 2m2−6m+5m−15=2m(m−3)+5(m−3)=(2m+5)(m−3)2m^2 - 6m + 5m - 15 = 2m(m - 3) + 5(m - 3) = (2m + 5)(m - 3).
Applying factoring by grouping.

Key Concept

Factoring Non-Monic Trinomials and Sign Analysis
Estimated Time:2m 30s
Question 10Question →

Which of the following represents the completely factored form of the algebraic expression 2x2−4x−302x^2 - 4x - 30?

Show answer & explanation

Answer: 2(x−5)(x+3)2(x - 5)(x + 3)

Answer

The completely factored form is 2(x−5)(x+3)2(x - 5)(x + 3).
To find the completely factored form, first factor out the greatest common factor (GCF), which is 22. This gives 2(x2−2x−15)2(x^2 - 2x - 15). Next, factor the trinomial inside the parentheses. You need two numbers that multiply to −15-15 and add up to −2-2. These numbers are −5-5 and 33. Substituting these into the binomial factors yields the final completely factored form: 2(x−5)(x+3)2(x - 5)(x + 3).

Step-by-Step Solution

1
Factor out the greatest common factor (GCF) from all terms in the expression 2x2−4x−302x^2 - 4x - 30.
The GCF is 22. Factoring it out leaves 2(x2−2x−15)2(x^2 - 2x - 15).
Identifying and extracting the GCF is always the first step in simplifying and completely factoring a polynomial.
2
Factor the remaining quadratic trinomial x2−2x−15x^2 - 2x - 15 by finding two integers that multiply to −15-15 and add to −2-2.
The two integers are −5-5 and 33.
For a trinomial in the form x2+bx+cx^2 + bx + c, the constants in the binomial factors must multiply to cc and add to bb.
3
Write the expression as the product of the GCF and the two binomial factors.
The fully factored expression is 2(x−5)(x+3)2(x - 5)(x + 3).
This combines all factored components into their most simplified, prime form.

Key Concept

Factoring out a greatest common factor (GCF) followed by factoring a quadratic trinomial.
Question 11Question →

What is the completely factored form of the polynomial 3x2−12x−363x^2 - 12x - 36?

Show answer & explanation

Answer: 3(x−6)(x+2)3(x - 6)(x + 2)

Answer

3(x−6)(x+2)3(x - 6)(x + 2)
To completely factor the polynomial, the greatest common factor must be extracted first. Factoring out the GCF of 3 yields 3(x2−4x−12)3(x^2 - 4x - 12). Next, factoring the trinomial x2−4x−12x^2 - 4x - 12 requires two integers that multiply to −12-12 and sum to −4-4. These integers are −6-6 and 22. Therefore, the completely factored form is 3(x−6)(x+2)3(x - 6)(x + 2).

Step-by-Step Solution

1
Factor out the greatest common factor (GCF) from all terms in the polynomial.
3(x2−4x−12)3(x^2 - 4x - 12)
Extracting the GCF simplifies the expression, making the quadratic trinomial easier to factor.
2
Find two numbers that multiply to the constant term (−12-12) and add to the coefficient of the middle term (−4-4).
The numbers are −6-6 and 22, because (−6)×2=−12(-6) \times 2 = -12 and (−6)+2=−4(-6) + 2 = -4.
These values are required to correctly split the middle term and factor the quadratic trinomial.
3
Write the completely factored form by combining the GCF and the two binomial factors.
3(x−6)(x+2)3(x - 6)(x + 2)
This represents the polynomial broken down into its simplest, irreducible algebraic factors.

Key Concept

Factoring polynomials by first extracting the Greatest Common Factor (GCF) and then factoring the resulting quadratic trinomial.
Question 12Question →

Complete the factored form of the given quadratic polynomial by determining the missing constant.

What number correctly fills in the blank?

Fill in the blanks below

$n^2 + 11n + 24 = (n + 3)(n + )$
Show answer & explanation

Answer

8
The polynomial n2+11n+24n^2 + 11n + 24 factors into two binomials. We are given one factor, (n+3)(n + 3). To find the missing value in the second factor (n+b)(n + b), we can use the property that the product of the constant terms in the binomials must equal the constant term of the trinomial. Therefore, 3×b=243 \times b = 24, which means b=8b = 8. We verify this by adding the constants: 3+8=113 + 8 = 11, which matches the coefficient of the middle term. Thus, the correct value for the blank is 8.

Step-by-Step Solution

1
Identify the relationship between the constant term of the polynomial and the constants in its binomial factors.
The product of the constants in the binomial factors must equal the constant term of the polynomial, 24.
When expanding (n+a)(n+b)(n + a)(n + b), the constant term is calculated as a×ba \times b.
2
Set up an equation to find the missing value.
3×missing value=243 \times \text{missing value} = 24.
One of the binomial factors is given as (n+3)(n + 3), so one constant is 3.
3
Solve for the missing value and verify with the middle term.
The missing value is 8, since 24÷3=824 \div 3 = 8. Checking the middle term, 3+8=113 + 8 = 11, which matches 11n11n.
This confirms that 8 satisfies both the multiplication to 24 and the addition to 11 requirements for factoring the trinomial.

Key Concept

Factoring quadratic trinomials of the form x^2 + bx + c.
Question 13Question →

The area of a rectangular board is given by the algebraic expression 4x2−8x−54x^2 - 8x - 5. Which of the following shows the dimensions of the board expressed as a product of two binomials?

Show answer & explanation

Answer: (2x+1)(2x−5)(2x + 1)(2x - 5)

Answer

The correct dimensions are given by the factored form (2x+1)(2x−5)(2x + 1)(2x - 5).
To factor 4x2−8x−54x^2 - 8x - 5, find two values that multiply to −20-20 (which is 4×−54 \times -5) and add to −8-8. These values are −10-10 and 22. Splitting the middle term gives 4x2−10x+2x−54x^2 - 10x + 2x - 5. Factoring by grouping gives 2x(2x−5)+1(2x−5)2x(2x - 5) + 1(2x - 5), which then factors out to (2x+1)(2x−5)(2x + 1)(2x - 5).

Step-by-Step Solution

1
Identify the coefficients for the quadratic expression ax2+bx+cax^2 + bx + c.
a=4a = 4, b=−8b = -8, and c=−5c = -5.
To factor a non-monic quadratic trinomial by grouping, we first need to determine the product of aa and cc.
2
Find two numbers that multiply to the product of aa and cc (−20-20) and add to bb (−8-8).
The two numbers are −10-10 and 22.
These numbers allow us to split the middle term (−8x-8x) into two parts that can be factored by grouping.
3
Rewrite the original expression by splitting the middle term.
4x2−10x+2x−54x^2 - 10x + 2x - 5
This sets up the expression for factoring by grouping.
4
Factor the greatest common factor (GCF) from the first two terms and the last two terms.
2x(2x−5)+1(2x−5)2x(2x - 5) + 1(2x - 5)
Extracting the GCF reveals a common binomial factor in both groups.
5
Factor out the common binomial factor.
(2x+1)(2x−5)(2x + 1)(2x - 5)
This provides the completely factored form of the original area expression.

Key Concept

Factoring Non-Monic Quadratic Trinomials
Question 14Question →

What is the completely factored form of the polynomial 3x2−3x−363x^2 - 3x - 36?

Show answer & explanation

Answer: 3(x−4)(x+3)3(x - 4)(x + 3)

Answer

The completely factored form is 3(x−4)(x+3)3(x - 4)(x + 3).
The correct answer accurately extracts the greatest common factor of 33 first, and then properly factors the resulting quadratic x2−x−12x^2 - x - 12 into (x−4)(x+3)(x - 4)(x + 3).

Step-by-Step Solution

1
Identify and factor out the greatest common factor (GCF) from all terms in the polynomial 3x2−3x−363x^2 - 3x - 36.
3(x2−x−12)3(x^2 - x - 12)
Factoring out the GCF simplifies the remaining quadratic expression, making it easier to factor.
2
Find two numbers that multiply to the constant term (−12-12) and add up to the coefficient of the middle term (−1-1).
The numbers are −4-4 and 33, because −4×3=−12-4 \times 3 = -12 and −4+3=−1-4 + 3 = -1.
These numbers are required to construct the binomial factors of the quadratic expression.
3
Write the final completely factored expression by combining the GCF and the two binomial factors.
3(x−4)(x+3)3(x - 4)(x + 3)
This represents the full decomposition of the original polynomial into its simplest multiplicative components.

Key Concept

Factoring out a greatest common factor (GCF) followed by factoring a basic quadratic trinomial.
Question 15Question →

Match each polynomial expression with its completely factored form.

Click a left item, then click its matching right item

Items

x2−25x^2 - 25
4x2+8x4x^2 + 8x
x2+7x+10x^2 + 7x + 10
x2−3x−10x^2 - 3x - 10

Matches

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Answer

The correct matches pair each polynomial with its fully factored equivalent using difference of squares, greatest common factor, and basic trinomial factoring rules.
Each correct pairing applies fundamental algebraic factoring rules: recognizing special binomial products like the difference of squares, pulling out the greatest common factor (GCF) from terms, and finding appropriate numerical binomial factors for quadratic trinomials.

Step-by-Step Solution

1
Analyze each polynomial to determine the applicable factoring method.
Identified difference of squares, GCF, and two basic quadratic trinomials.
Different polynomial structures require specific factoring techniques.
2
Factor the difference of squares: x2−25x^2 - 25.
Matched with (x−5)(x+5)(x - 5)(x + 5).
The pattern a2−b2a^2 - b^2 factors into (a−b)(a+b)(a - b)(a + b).
3
Factor out the greatest common factor for 4x2+8x4x^2 + 8x.
Matched with 4x(x+2)4x(x + 2).
Both terms share a maximum common factor of 4x4x.
4
Factor the trinomial x2+7x+10x^2 + 7x + 10.
Matched with (x+2)(x+5)(x + 2)(x + 5).
The numbers 22 and 55 multiply to positive 1010 and add to positive 77.
5
Factor the trinomial x2−3x−10x^2 - 3x - 10.
Matched with (x−5)(x+2)(x - 5)(x + 2).
The numbers −5-5 and 22 multiply to negative 1010 and add to negative 33.

Key Concept

Factoring Polynomials
Question 16Question →

Which of the following represents the completely factored form of the algebraic expression 3x2−273x^2 - 27?

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Answer: 3(x−3)(x+3)3(x - 3)(x + 3)

Answer

The completely factored form is 3(x−3)(x+3)3(x - 3)(x + 3).
The expression is completely factored by first extracting the greatest common factor of 3 to get 3(x2−9)3(x^2 - 9), and then factoring the resulting difference of squares into (x−3)(x+3)(x - 3)(x + 3).

Step-by-Step Solution

1
Identify and factor out the greatest common factor (GCF) from all terms in the expression.
3(x2−9)3(x^2 - 9)
Both terms share a common factor of 3. Factoring this out simplifies the remaining polynomial.
2
Examine the expression inside the parentheses to see if it can be factored further.
x2−9x^2 - 9 is a difference of squares.
The expression x2−9x^2 - 9 follows the pattern a2−b2a^2 - b^2, where a=xa = x and b=3b = 3.
3
Factor the difference of squares using the standard algebraic pattern (a−b)(a+b)(a - b)(a + b).
(x−3)(x+3)(x - 3)(x + 3)
Expanding (x−3)(x+3)(x - 3)(x + 3) yields x2+3x−3x−9x^2 + 3x - 3x - 9, which simplifies back to x2−9x^2 - 9.
4
Combine the GCF and the factored binomials to write the final completely factored form.
3(x−3)(x+3)3(x - 3)(x + 3)
An expression is only completely factored when all of its factors are prime (cannot be factored further).

Key Concept

Factoring by finding a Greatest Common Factor (GCF) followed by factoring a difference of squares.
Question 17Question →

During an analysis of a parabolic trajectory, an engineer must factor the polynomial 4x2−4x−484x^2 - 4x - 48 completely to determine its zero points. What is the correctly factored form of this polynomial?

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Answer: 4(x−4)(x+3)4(x - 4)(x + 3)

Answer

The completely factored form of the polynomial is 4(x−4)(x+3)4(x - 4)(x + 3).
To factor completely, first factor out the GCF of 4 to get 4(x2−x−12)4(x^2 - x - 12). Then, factor the resulting trinomial by finding two numbers that multiply to -12 and add to -1. Those numbers are -4 and 3, yielding the final form 4(x−4)(x+3)4(x - 4)(x + 3).

Step-by-Step Solution

1
Extract the greatest common factor (GCF) from all terms in the polynomial 4x2−4x−484x^2 - 4x - 48.
4(x2−x−12)4(x^2 - x - 12)
Factoring out the GCF simplifies the polynomial and makes the remaining quadratic expression easier to factor.
2
Find two integers that multiply to the constant term (-12) and add up to the coefficient of the middle term (-1).
The two integers are -4 and +3.
These integers provide the constants needed for the two binomial factors of the quadratic trinomial.
3
Write the final expression by combining the GCF with the factored binomials.
4(x−4)(x+3)4(x - 4)(x + 3)
This shows the polynomial broken down into its prime, indivisible factors.

Key Concept

Factoring out a Greatest Common Factor (GCF) before factoring a quadratic trinomial.
Question 18Question →

When the algebraic expression 12y3+12y2−45y12y^3 + 12y^2 - 45y is factored completely, which of the following expressions is produced?

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Answer: 3y(2y−3)(2y+5)3y(2y - 3)(2y + 5)

Answer

The completely factored form is 3y(2y−3)(2y+5)3y(2y - 3)(2y + 5).
The correct answer accurately extracts the full GCF of 3y3y and correctly factors the remaining trinomial into (2y−3)(2y+5)(2y - 3)(2y + 5). Expanding this result confirms it equals 12y3+12y2−45y12y^3 + 12y^2 - 45y.

Step-by-Step Solution

1
Identify and extract the Greatest Common Factor (GCF) from all terms in 12y3+12y2−45y12y^3 + 12y^2 - 45y.
The GCF is 3y3y. Factoring it out leaves 3y(4y2+4y−15)3y(4y^2 + 4y - 15).
Extracting the GCF first simplifies the polynomial and is a required step for complete factorization.
2
Factor the remaining non-monic trinomial, 4y2+4y−154y^2 + 4y - 15, by finding two numbers that multiply to a×ca \times c and add to bb.
We need numbers that multiply to (4)(−15)=−60(4)(-15) = -60 and add to 44. The numbers are 1010 and −6-6.
This prepares the trinomial for factoring by grouping.
3
Rewrite the middle term (4y4y) using these two numbers and factor by grouping.
4y2+10y−6y−15=2y(2y+5)−3(2y+5)=(2y−3)(2y+5)4y^2 + 10y - 6y - 15 = 2y(2y + 5) - 3(2y + 5) = (2y - 3)(2y + 5).
Grouping extracts common binomial factors, breaking down the quadratic into its simplest linear binomials.
4
Combine the original GCF with the newly factored binomials.
3y(2y−3)(2y+5)3y(2y - 3)(2y + 5)
To represent the full and complete factorization of the original polynomial.

Key Concept

Factoring polynomials by first extracting the GCF and subsequently using the ac-method (or grouping) to factor a non-monic quadratic trinomial.
Question 19Question →

The polynomial x2+6x−16x^2 + 6x - 16 can be factored as (x−2)(x+k)(x - 2)(x + k). What is the value of the constant kk?

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Answer: 8; +8; eight

Answer

8
The constant kk must be 8 because in the factored form (x−2)(x+k)(x - 2)(x + k), the product of -2 and kk must equal the constant term of the polynomial, which is -16. Since −2×8=−16-2 \times 8 = -16 and the sum −2+8=6-2 + 8 = 6 gives the correct middle coefficient, the value of kk is exactly 8.

Step-by-Step Solution

1
Analyze the relationship between the expanded polynomial and its factored form.
The expanded form is x2+6x−16x^2 + 6x - 16 and the factored form is (x−2)(x+k)(x - 2)(x + k).
This sets up the equation to solve for the missing constant kk.
2
Use the property of multiplying binomials to find the constant term.
The product of the constants in the binomials must equal the constant term of the polynomial: −2×k=−16-2 \times k = -16.
When expanding (x+a)(x+b)(x + a)(x + b), the constant term is always equal to a×ba \times b.
3
Solve the equation for kk.
k=−16÷(−2)=8k = -16 \div (-2) = 8.
Dividing both sides by -2 isolates kk.
4
Verify the middle term coefficient with the found value of kk.
−2+8=6-2 + 8 = 6, which matches the middle term 6x6x.
Checking the sum of the constants ensures the entire factorization is correct.

Key Concept

Factoring quadratic trinomials by finding two numbers that multiply to the constant term and add to the middle coefficient.
Question 20Question →

When the algebraic expression 5x2−13x−65x^2 - 13x - 6 is factored completely, it can be written in the form (5x+a)(x−b)(5x + a)(x - b), where aa and bb are positive integers. What is the value of a+ba + b?

Show answer & explanation

Answer: 5; five

Answer

5
Factoring the quadratic 5x2−13x−65x^2 - 13x - 6 yields (5x+2)(x−3)(5x + 2)(x - 3). By comparing this to the target format (5x+a)(x−b)(5x + a)(x - b), we can identify that a=2a = 2 and b=3b = 3. Adding these together gives the final answer of 5.

Step-by-Step Solution

1
Determine the factoring method for the non-monic quadratic expression 5x2−13x−65x^2 - 13x - 6.
Since the leading coefficient is 5, look for two numbers that multiply to (5)(−6)=−30(5)(-6) = -30 and add to the middle coefficient, −13-13.
This sets up the factor-by-grouping method.
2
Find the two numbers.
The numbers are −15-15 and 22.
(−15)×2=−30(-15) \times 2 = -30 and −15+2=−13-15 + 2 = -13.
3
Rewrite the middle term using these two numbers and factor by grouping.
5x2−15x+2x−6=5x(x−3)+2(x−3)=(5x+2)(x−3)5x^2 - 15x + 2x - 6 = 5x(x - 3) + 2(x - 3) = (5x + 2)(x - 3).
Grouping common terms extracts the binomial factors.
4
Match the factored form to the given format (5x+a)(x−b)(5x + a)(x - b).
Comparing (5x+2)(x−3)(5x + 2)(x - 3) to (5x+a)(x−b)(5x + a)(x - b) reveals that a=2a = 2 and b=3b = 3.
Both aa and bb correspond exactly to the positive integer components of the binomials.
5
Calculate the final required sum, a+ba + b.
2+3=52 + 3 = 5.
The question specifically asks for the sum of the parameters.

Key Concept

Factoring non-monic quadratic trinomials
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