Examkin

Algebra

282 questions

All types282Multiple choice147True / False7Numeric33Matching33Fill in the blank39Short answer23
Question 41Question →

Which of the following expressions is equivalent to 6x2−15x−366x^2 - 15x - 36 and is factored completely?

Show answer & explanation

Answer: 3(2x+3)(x−4)3(2x + 3)(x - 4)

Answer

The completely factored expression is 3(2x+3)(x−4)3(2x + 3)(x - 4).
The correct answer identifies the GCF of 3 and correctly factors the remaining quadratic 2x2−5x−122x^2 - 5x - 12 into (2x+3)(x−4)(2x + 3)(x - 4). Expanding this result confirms it equals 6x2−15x−366x^2 - 15x - 36 and no further factors can be extracted.

Step-by-Step Solution

1
Identify and factor out the greatest common factor (GCF) from all three terms.
3(2x2−5x−12)3(2x^2 - 5x - 12)
The coefficients 6, -15, and -36 are all divisible by 3.
2
Find two numbers that multiply to acac (where a=2a=2 and c=−12c=-12, so 2×−12=−242 \times -12 = -24) and add to the middle coefficient (b=−5b=-5).
The numbers are −8-8 and 33.
(−8)×3=−24(-8) \times 3 = -24 and (−8)+3=−5(-8) + 3 = -5.
3
Rewrite the trinomial by splitting the middle term using these two numbers.
3(2x2−8x+3x−12)3(2x^2 - 8x + 3x - 12)
This sets up the polynomial for factoring by grouping.
4
Factor the trinomial by grouping.
3[2x(x−4)+3(x−4)]=3(2x+3)(x−4)3[2x(x - 4) + 3(x - 4)] = 3(2x + 3)(x - 4)
Factoring out (x−4)(x - 4) from both groups provides the final completely factored form.

Key Concept

Factoring out a GCF followed by factoring a non-monic trinomial.
Question 42Question →

Which of the following represents the completely factored form of the algebraic expression 30v3−35v2−15v30v^3 - 35v^2 - 15v?

Show answer & explanation

Answer: 5v(3v+1)(2v−3)5v(3v + 1)(2v - 3)

Answer

The completely factored form is 5v(3v+1)(2v−3)5v(3v + 1)(2v - 3).
The correct answer accurately extracts the Greatest Common Factor of 5v from the original expression, leaving the non-monic trinomial 6v^2 - 7v - 3. Factoring this remaining trinomial by grouping correctly yields the binomials (3v + 1) and (2v - 3). Expanding these prime factors back confirms they produce the exact original expression.

Step-by-Step Solution

1
Identify and factor out the greatest common factor (GCF) from all three terms in the polynomial 30v3−35v2−15v30v^3 - 35v^2 - 15v.
5v(6v2−7v−3)5v(6v^2 - 7v - 3)
Factoring out the GCF simplifies the remaining polynomial and is a necessary first step for complete factorization.
2
Analyze the remaining quadratic trinomial 6v2−7v−36v^2 - 7v - 3 to factor it by grouping. Find two numbers that multiply to a×ca \times c (6×−3=−186 \times -3 = -18) and add to bb (−7-7).
The two numbers are −9-9 and 22.
These numbers allow us to split the middle term (−7v-7v) into two separate terms to enable factoring by grouping.
3
Rewrite the trinomial by splitting the middle term using the numbers found, then factor by grouping.
6v2−9v+2v−3=3v(2v−3)+1(2v−3)=(3v+1)(2v−3)6v^2 - 9v + 2v - 3 = 3v(2v - 3) + 1(2v - 3) = (3v + 1)(2v - 3)
Grouping the terms reveals the common binomial factor of (2v−3)(2v - 3).
4
Combine the GCF extracted in step 1 with the binomial factors found in step 3.
5v(3v+1)(2v−3)5v(3v + 1)(2v - 3)
This represents the original expression as a product of its simplest irreducible factors.

Key Concept

Factoring a polynomial by first extracting the greatest common factor (GCF), followed by factoring a non-monic quadratic trinomial using grouping.
Question 43Question →

Decompose the quadratic trinomial 6x2−11x−106x^2 - 11x - 10 into a product of two binomial factors structured as (ax−b)(cx+d)(ax - b)(cx + d). Assume all parameters (aa, bb, cc, and dd) are strictly positive integers. To ensure a unique arrangement, the leading coefficient aa must be strictly less than cc. Provide the values of the four parameters.

Fill in the blanks below

a=a = , b=b = , c=c = , and d=d = .
Show answer & explanation

Answer

The parameters are a=2a = 2, b=5b = 5, c=3c = 3, and d=2d = 2.
Factoring the polynomial 6x2−11x−106x^2 - 11x - 10 using the AC method produces (2x−5)(3x+2)(2x - 5)(3x + 2). Matching these two factors against the template (ax−b)(cx+d)(ax - b)(cx + d) under the rule that a<ca < c (since 2<32 < 3) perfectly determines the positive integers as a=2a = 2, b=5b = 5, c=3c = 3, and d=2d = 2.

Step-by-Step Solution

1
Identify the factoring technique for a non-monic quadratic trinomial in the form Ax2+Bx+CAx^2 + Bx + C.
The AC method (factoring by grouping) is required because the leading coefficient A=6A = 6 is greater than 1.
Directly finding factors of the constant CC that sum to BB only works when A=1A = 1.
2
Multiply the leading coefficient AA by the constant term CC.
6×(−10)=−606 \times (-10) = -60.
This establishes the target product needed to properly split the middle term for grouping.
3
Find two integers that multiply to −60-60 and add to the middle coefficient B=−11B = -11.
The correct pair of integers is −15-15 and 44.
(−15)×4=−60(-15) \times 4 = -60 and −15+4=−11-15 + 4 = -11.
4
Rewrite the middle term using these two integers and factor the resulting four-term expression by grouping.
6x2−15x+4x−10=3x(2x−5)+2(2x−5)=(2x−5)(3x+2)6x^2 - 15x + 4x - 10 = 3x(2x - 5) + 2(2x - 5) = (2x - 5)(3x + 2).
Grouping the first two terms and the last two terms reveals a common binomial factor of (2x−5)(2x - 5).
5
Match the factored expression to the required template (ax−b)(cx+d)(ax - b)(cx + d) while adhering to the condition a<ca < c.
The factors are (2x−5)(2x - 5) and (3x+2)(3x + 2). Since 2<32 < 3, (ax−b)(ax - b) corresponds exactly to (2x−5)(2x - 5), yielding a=2a=2 and b=5b=5. Consequently, (cx+d)(cx + d) corresponds to (3x+2)(3x + 2), yielding c=3c=3 and d=2d=2.
This guarantees a unique, correctly-ordered set of strictly positive parameters.

Key Concept

Factoring non-monic quadratic trinomials via grouping
Question 44Question →

Complete the factorization of the perfect square trinomial below by filling in the missing number.

Fill in the blanks below

$x^2 + 14x + 49 = (x + )^2$
Show answer & explanation

Answer

7
A perfect square trinomial takes the general form x2+2bx+b2=(x+b)2x^2 + 2bx + b^2 = (x + b)^2. In the given expression x2+14x+49x^2 + 14x + 49, the constant term is 4949, which is 727^2. The middle term is 14x14x, which is equal to 2×7×x2 \times 7 \times x. Therefore, the expression factors directly as (x+7)2(x + 7)^2, meaning the missing number in the blank is 7.

Step-by-Step Solution

1
Recognize the polynomial form
The expression x2+14x+49x^2 + 14x + 49 matches the structure of a perfect square trinomial: a2+2ab+b2a^2 + 2ab + b^2.
The first term is a perfect square (x2x^2), and the last term is a perfect square (49=7249 = 7^2).
2
Verify the middle term
Checking the middle term: 2×x×7=14x2 \times x \times 7 = 14x. This matches the middle term of the given expression.
Confirming the middle term proves the expression can be factored exactly as (a+b)2(a + b)^2.
3
Determine the missing value
The expression factors as (x+7)2(x + 7)^2.
Using the formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2 where a=xa = x and b=7b = 7.

Key Concept

Factoring perfect square trinomials
Question 45Question →

An engineer needs to simplify a calculation involving the volume of a component, modeled by the polynomial 4y3+12y2−72y4y^3 + 12y^2 - 72y. What is the fully factored equivalent of this polynomial?

Show answer & explanation

Answer: 4y(y+6)(y−3)4y(y + 6)(y - 3)

Answer

4y(y+6)(y−3)4y(y + 6)(y - 3)
To fully factor the given expression, first extract the greatest common factor (GCF). All terms share a common factor of 4y4y, yielding 4y(y2+3y−18)4y(y^2 + 3y - 18). Next, factor the resulting quadratic trinomial y2+3y−18y^2 + 3y - 18. The two numbers that multiply to −18-18 and sum to 33 are 66 and −3-3. This gives the binomial factors (y+6)(y + 6) and (y−3)(y - 3). Combining these with the GCF results in the completely factored form 4y(y+6)(y−3)4y(y + 6)(y - 3).

Step-by-Step Solution

1
Identify and extract the greatest common factor (GCF) from all terms in the polynomial 4y3+12y2−72y4y^3 + 12y^2 - 72y.
4y(y2+3y−18)4y(y^2 + 3y - 18)
Factoring out the GCF simplifies the remaining polynomial, making it easier to factor completely.
2
Factor the remaining quadratic trinomial y2+3y−18y^2 + 3y - 18 by finding two integers that multiply to −18-18 and add to 33.
(y+6)(y−3)(y + 6)(y - 3)
The integers 66 and −3-3 satisfy both conditions: 6×(−3)=−186 \times (-3) = -18 and 6+(−3)=36 + (-3) = 3.
3
Combine the GCF with the factored binomials to state the final expression.
4y(y+6)(y−3)4y(y + 6)(y - 3)
This represents the original polynomial decomposed entirely into its lowest degree factors.

Key Concept

Factoring Polynomials with a GCF
Estimated Time:1m 0s
Question 46Question →

Pair each algebraic expression with its completely factored equivalent.

Click a left item, then click its matching right item

Items

4y2−16y4y^2 - 16y
y2−16y^2 - 16
y2−8y+16y^2 - 8y + 16
4y+164y + 16

Matches

Show answer & explanation

Answer

The correct pairings are: 4y2−16y4y^2 - 16y matches with 4y(y−4)4y(y - 4); y2−16y^2 - 16 matches with (y−4)(y+4)(y - 4)(y + 4); y2−8y+16y^2 - 8y + 16 matches with (y−4)2(y - 4)^2; and 4y+164y + 16 matches with 4(y+4)4(y + 4).
Each polynomial requires a distinct factoring method. Identifying the correct structure—such as GCF, difference of squares, or perfect square trinomial—leads directly to the completely factored form.

Step-by-Step Solution

1
Factor 4y2−16y4y^2 - 16y
4y(y−4)4y(y - 4)
The terms share a greatest common factor of 4y4y.
2
Factor y2−16y^2 - 16
(y−4)(y+4)(y - 4)(y + 4)
It follows the difference of squares pattern a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b).
3
Factor y2−8y+16y^2 - 8y + 16
(y−4)2(y - 4)^2
It is a perfect square trinomial, as (−4)×(−4)=16(-4) \times (-4) = 16 and (−4)+(−4)=−8(-4) + (-4) = -8.
4
Factor 4y+164y + 16
4(y+4)4(y + 4)
The terms share a greatest common factor of 44.

Key Concept

Factoring polynomials using greatest common factors, difference of squares, and perfect square trinomial patterns.
Question 47Question →

Match each polynomial expression to its completely factored form.

Click a left item, then click its matching right item

Items

x2−16x^2 - 16
x2+6x+8x^2 + 6x + 8
2x2−12x+182x^2 - 12x + 18
3x2−5x−23x^2 - 5x - 2

Matches

Show answer & explanation

Answer

The correct matches pair each polynomial with its mathematically equivalent fully factored expression.
Each polynomial requires identifying its specific factoring archetype: the difference of two squares, a standard quadratic trinomial, GCF extraction followed by recognizing a perfect square trinomial, or factoring a non-monic trinomial.

Step-by-Step Solution

1
Analyze the binomial x2−16x^2 - 16.
Matches to (x−4)(x+4)(x - 4)(x + 4).
It represents a difference of perfect squares.
2
Analyze the trinomial x2+6x+8x^2 + 6x + 8.
Matches to (x+2)(x+4)(x + 2)(x + 4).
The constants 2 and 4 have a product of 8 and a sum of 6.
3
Analyze the polynomial 2x2−12x+182x^2 - 12x + 18.
Matches to 2(x−3)22(x - 3)^2.
Factoring out the common factor of 2 leaves x2−6x+9x^2 - 6x + 9, which is the square of (x−3)(x - 3).
4
Analyze the polynomial 3x2−5x−23x^2 - 5x - 2.
Matches to (3x+1)(x−2)(3x + 1)(x - 2).
Multiplying the binomials (3x+1)(3x + 1) and (x−2)(x - 2) confirms it expands to the original expression.

Key Concept

Factoring different archetypes of polynomials, including the difference of squares, monic trinomials, non-monic trinomials, and extracting the Greatest Common Factor (GCF).
Question 48Question →
When the polynomial expression (3x−2y)(Ax2+4xy−5y2)(3x - 2y)(Ax^2 + 4xy - 5y^2) is fully expanded and simplified, the coefficient of the x2yx^2y term is −4-4. What is the value of the constant AA?
Show answer & explanation

Answer: 88

Answer

The value of the constant AA is 88.
To find the coefficient of the x2yx^2y term, we do not need to expand the entire expression. We only need to find the specific multiplications that result in an x2yx^2y term. Multiplying the xx term from the first polynomial by the xyxy term from the second gives 3x×4xy=12x2y3x \times 4xy = 12x^2y. Multiplying the yy term from the first polynomial by the x2x^2 term from the second gives −2y×Ax2=−2Ax2y-2y \times Ax^2 = -2Ax^2y. Combining these gives a total x2yx^2y coefficient of (12−2A)(12 - 2A). Since the problem states this coefficient is −4-4, we set up the equation 12−2A=−412 - 2A = -4. Subtracting 1212 from both sides gives −2A=−16-2A = -16. Dividing by −2-2 gives A=8A = 8.

Step-by-Step Solution

1
Identify the products that result in an x2yx^2y term when expanding the polynomials.
The relevant terms are (3x)(4xy)(3x)(4xy) and (−2y)(Ax2)(-2y)(Ax^2).
Only these specific combinations of terms will produce the exact variable configuration x2yx^2y.
2
Multiply these specific terms to find their coefficients.
(3x)(4xy)=12x2y(3x)(4xy) = 12x^2y and (−2y)(Ax2)=−2Ax2y(-2y)(Ax^2) = -2Ax^2y.
This isolates the parts of the expansion that contribute to the final x2yx^2y coefficient.
3
Combine the coefficients and set them equal to the given total coefficient of −4-4.
12−2A=−412 - 2A = -4
The algebraic sum of the individual x2yx^2y coefficients must equal the final simplified coefficient provided in the problem.
4
Solve the linear equation for AA.
−2A=−4−12⇒−2A=−16⇒A=8-2A = -4 - 12 \Rightarrow -2A = -16 \Rightarrow A = 8
Standard algebraic isolation of the variable AA gives the final answer.

Key Concept

Identifying specific terms in polynomial multiplication and solving for an unknown coefficient.
Estimated Time:1m 30s
Question 49Question →

To optimize a packaging design, a technician models the surface area using the algebraic expression 18xy2−24xy−10x18xy^2 - 24xy - 10x. What is the fully factored equivalent of this expression?

Show answer & explanation

Answer: 2x(3y−5)(3y+1)2x(3y - 5)(3y + 1)

Answer

The fully factored equivalent is 2x(3y−5)(3y+1)2x(3y - 5)(3y + 1).
The expression is fully factored by first extracting the greatest common factor of 2x2x, resulting in 2x(9y2−12y−5)2x(9y^2 - 12y - 5). Then, using the ac-method, the remaining non-monic quadratic trinomial is factored into (3y−5)(3y+1)(3y - 5)(3y + 1), yielding the final correct product.

Step-by-Step Solution

1
Identify and extract the greatest common factor (GCF) from all three terms.
2x(9y2−12y−5)2x(9y^2 - 12y - 5)
Extracting the GCF simplifies the polynomial and prepares the remaining quadratic trinomial for factoring.
2
Identify the components of the remaining quadratic trinomial ay2+by+cay^2 + by + c.
Here, a=9a = 9, b=−12b = -12, and c=−5c = -5.
This establishes the values needed to use the ac-method for factoring non-monic quadratics.
3
Find two numbers that multiply to acac and add to bb.
ac=9×−5=−45ac = 9 \times -5 = -45. The two numbers that multiply to −45-45 and add to −12-12 are −15-15 and 33.
These numbers allow the middle term to be split, enabling factoring by grouping.
4
Rewrite the middle term and factor the trinomial by grouping.
9y2−15y+3y−5=3y(3y−5)+1(3y−5)=(3y−5)(3y+1)9y^2 - 15y + 3y - 5 = 3y(3y - 5) + 1(3y - 5) = (3y - 5)(3y + 1)
This completes the factorization of the trinomial portion of the expression.
5
Combine the originally extracted GCF with the factored binomials.
2x(3y−5)(3y+1)2x(3y - 5)(3y + 1)
This yields the complete and fully factored form of the original polynomial.

Key Concept

Factoring Polynomials with a GCF
Question 50Question →

If 3x+5y=113x + 5y = 11 and 4x+9y=34x + 9y = 3, what is the value of the expression x+2yx + 2y?

Show answer & explanation

Answer: 2

Answer

2
The correct answer is 2. To solve the system, use the elimination method by multiplying the equations to create opposite or matching coefficients. Multiplying the first equation by 4 yields 12x+20y=4412x + 20y = 44. Multiplying the second equation by 3 yields 12x+27y=912x + 27y = 9. Subtracting the first equation from the second gives 7y=−357y = -35, so y=−5y = -5. Substituting y=−5y = -5 into the first original equation gives 3x−25=113x - 25 = 11, which simplifies to 3x=363x = 36, or x=12x = 12. Finally, substituting these values into the expression x+2yx + 2y gives 12+2(−5)=12−10=212 + 2(-5) = 12 - 10 = 2.

Step-by-Step Solution

1
Multiply the first equation by 4 and the second equation by 3 to prepare for elimination.
12x+20y=4412x + 20y = 44 and 12x+27y=912x + 27y = 9
Creating matching coefficients for xx allows us to eliminate it by subtracting the equations.
2
Subtract the first new equation from the second new equation.
(12x+27y)−(12x+20y)=9−44⇒7y=−35(12x + 27y) - (12x + 20y) = 9 - 44 \Rightarrow 7y = -35
This eliminates xx and leaves a simple equation with only yy.
3
Divide by 7 to solve for yy.
y=−5y = -5
Isolating yy gives its numerical value.
4
Substitute y=−5y = -5 back into the original first equation (3x+5y=113x + 5y = 11) and solve for xx.
3x+5(−5)=11⇒3x−25=11⇒3x=36⇒x=123x + 5(-5) = 11 \Rightarrow 3x - 25 = 11 \Rightarrow 3x = 36 \Rightarrow x = 12
This finds the corresponding value for xx.
5
Substitute x=12x = 12 and y=−5y = -5 into the requested expression x+2yx + 2y.
12+2(−5)=12−10=212 + 2(-5) = 12 - 10 = 2
The question asks for the value of this specific expression, not just the ordered pair.

Key Concept

Solving systems of linear equations using the elimination method and evaluating algebraic expressions.
Estimated Time:2m 0s
Question 51Question →

A rectangular garden has an area given by the expression 4n2−4n−244n^2 - 4n - 24. Which of the following represents the completely factored form of this area expression?

Show answer & explanation

Answer: 4(n−3)(n+2)4(n - 3)(n + 2)

Answer

The completely factored form is 4(n−3)(n+2)4(n - 3)(n + 2).
To factor the expression completely, first remove the greatest common factor of 4 from all terms, which gives 4(n2−n−6)4(n^2 - n - 6). Then, factor the trinomial inside the parentheses. We need two numbers that multiply to -6 and add to -1, which are -3 and 2. This gives the completely factored form 4(n−3)(n+2)4(n - 3)(n + 2).

Step-by-Step Solution

1
Identify and factor out the greatest common factor (GCF) from all terms in the polynomial 4n2−4n−244n^2 - 4n - 24.
4(n2−n−6)4(n^2 - n - 6)
Factoring out the GCF simplifies the polynomial and is the required first step for complete factorization.
2
Factor the remaining quadratic trinomial n2−n−6n^2 - n - 6 by finding two numbers that multiply to -6 and add to -1.
The numbers are -3 and 2, yielding (n−3)(n+2)(n - 3)(n + 2).
This breaks down the trinomial into its constituent binomial factors.
3
Combine the GCF and the binomial factors to write the final expression.
4(n−3)(n+2)4(n - 3)(n + 2)
This represents the full, completely factored form of the original area expression.

Key Concept

Factoring out a GCF followed by factoring a quadratic trinomial
Question 52Question →
When analyzing algebraic interactions, it is often necessary to determine the exact coefficient of a specific term. Find the fully expanded and simplified form of the polynomial product (3a−4b)2(2a+b)(3a - 4b)^2(2a + b). What is the numerical coefficient of the a2ba^2b term?
Show answer & explanation

Answer: -39

Answer

-39
To find the coefficient of the a2ba^2b term, first expand the squared binomial (3a−4b)2(3a - 4b)^2 to get the trinomial 9a2−24ab+16b29a^2 - 24ab + 16b^2. Next, multiply this trinomial by (2a+b)(2a + b). The specific terms that result in a2ba^2b are the product of 9a29a^2 and bb (which yields 9a2b9a^2b) and the product of −24ab-24ab and 2a2a (which yields −48a2b-48a^2b). Combining these like terms gives 9a2b−48a2b=−39a2b9a^2b - 48a^2b = -39a^2b. Therefore, the numerical coefficient is -39.

Step-by-Step Solution

1
Expand the squared binomial (3a−4b)2(3a - 4b)^2.
9a2−24ab+16b29a^2 - 24ab + 16b^2
Order of operations requires resolving the exponent before multiplying by the binomial (2a+b)(2a + b).
2
Set up the multiplication of the resulting trinomial and the binomial.
(9a2−24ab+16b2)(2a+b)(9a^2 - 24ab + 16b^2)(2a + b)
This prepares the expression for distributing every term in the trinomial to every term in the binomial.
3
Identify and calculate only the partial products that result in an a2ba^2b term.
(9a2)(b)=9a2b(9a^2)(b) = 9a^2b and (−24ab)(2a)=−48a2b(-24ab)(2a) = -48a^2b
This is a more efficient method than fully expanding the entire polynomial if only a specific coefficient is needed.
4
Combine the identified like terms.
9a2b−48a2b=−39a2b9a^2b - 48a^2b = -39a^2b
Combining these terms gives the final, simplified coefficient for the a2ba^2b term.

Key Concept

Multiplying polynomials, expanding binomial squares, and combining like terms.
Question 53Question →

Which of the following expressions represents the completely factored form of the polynomial 3k3−15k2−42k3k^3 - 15k^2 - 42k?

Show answer & explanation

Answer: 3k(k−7)(k+2)3k(k - 7)(k + 2)

Answer

The completely factored form is 3k(k−7)(k+2)3k(k - 7)(k + 2).
The polynomial is completely factored by first extracting the Greatest Common Factor, 3k3k, which simplifies the expression to 3k(k2−5k−14)3k(k^2 - 5k - 14). Next, the remaining trinomial is factored by finding two numbers that multiply to −14-14 and add to −5-5. These numbers are −7-7 and 22. Combining these gives the final fully factored form of 3k(k−7)(k+2)3k(k - 7)(k + 2).

Step-by-Step Solution

1
Identify and factor out the Greatest Common Factor (GCF) from all three terms of the polynomial.
The GCF of 3k33k^3, −15k2-15k^2, and −42k-42k is 3k3k. Factoring this out leaves 3k(k2−5k−14)3k(k^2 - 5k - 14).
Factoring out the GCF simplifies the remaining polynomial and is the required first step for complete factorization.
2
Factor the remaining quadratic trinomial, k2−5k−14k^2 - 5k - 14.
Find two integers that multiply to −14-14 (the constant term) and add to −5-5 (the coefficient of the middle term). These integers are −7-7 and 22. This yields the binomial factors (k−7)(k - 7) and (k+2)(k + 2).
These binomials represent the linear factors of the quadratic expression.
3
Combine the GCF and the binomial factors to write the fully factored expression.
3k(k−7)(k+2)3k(k - 7)(k + 2)
This represents the original polynomial as a product of its simplest prime factors.

Key Concept

Factoring polynomials by first extracting the Greatest Common Factor (GCF) and then factoring the resulting quadratic trinomial.
Estimated Time:1m 0s
Question 54Question →

The algebraic expression 5x2+35x5x^2 + 35x can be completely factored into the form 5x(x+c)5x(x + c). What is the value of the constant cc?

Show answer & explanation

Answer: 7; +7; seven; Seven

Answer

7
By extracting the greatest common factor of 5x5x from 5x2+35x5x^2 + 35x, we divide both terms by 5x5x. This leaves (x+7)(x + 7) inside the parentheses. Comparing this to the given structure 5x(x+c)5x(x + c), it is clear that cc must be 7.

Step-by-Step Solution

1
Identify the greatest common factor (GCF) of the terms 5x25x^2 and 35x35x.
The GCF is 5x5x.
Both terms are divisible by 5 and share a minimum of one xx.
2
Divide each term of the original expression by the GCF to find the binomial factor.
5x25x=x\frac{5x^2}{5x} = x and 35x5x=7\frac{35x}{5x} = 7.
Factoring out the GCF is the reverse of distributing it.
3
Write the factored form and compare it to the given form 5x(x+c)5x(x + c).
The factored form is 5x(x+7)5x(x + 7). Therefore, comparing it to 5x(x+c)5x(x + c), c=7c = 7.
Matching corresponding terms reveals the value of the unknown constant.

Key Concept

Factoring the Greatest Common Factor (GCF)
Question 55Question →

When the polynomial 5x2−205x^2 - 20 is factored completely into the form k(x−c)(x+c)k(x - c)(x + c), where kk and cc are positive integers, what is the value of the sum k+ck + c?

Show answer & explanation

Answer: 7; seven

Answer

7
The polynomial 5x2−205x^2 - 20 can be factored completely by first pulling out the greatest common factor, 55, leaving 5(x2−4)5(x^2 - 4). The expression inside the parentheses is a difference of squares and factors into (x−2)(x+2)(x - 2)(x + 2). This makes the completely factored form 5(x−2)(x+2)5(x - 2)(x + 2). Comparing this to the form k(x−c)(x+c)k(x - c)(x + c), we see that k=5k = 5 and c=2c = 2. Their sum is 5+2=75 + 2 = 7.

Step-by-Step Solution

1
Factor out the greatest common factor (GCF) from the polynomial.
5x2−20=5(x2−4)5x^2 - 20 = 5(x^2 - 4)
Both terms share a common numerical factor of 55. Factoring it out simplifies the remaining polynomial.
2
Factor the remaining binomial as a difference of squares.
5(x2−4)=5(x−2)(x+2)5(x^2 - 4) = 5(x - 2)(x + 2)
The binomial x2−4x^2 - 4 follows the pattern a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b), where a=xa = x and b=2b = 2.
3
Identify the values of kk and cc from the given form.
k=5k = 5 and c=2c = 2
By matching 5(x−2)(x+2)5(x - 2)(x + 2) to the target form k(x−c)(x+c)k(x - c)(x + c), the constant multiplier is 55 and the positive integer in the binomial is 22.
4
Calculate the sum of kk and cc.
5+2=75 + 2 = 7
The question asks for the final value of the sum k+ck + c.

Key Concept

Factoring polynomials using the greatest common factor and the difference of squares.
Question 56Question →

Given the system of equations below, what is the value of the expression x−2yx - 2y?

5x−13=4y5x - 13 = 4y
2x+y=132x + y = 13
Show answer & explanation

Answer: -1

Answer

The correct value of the expression is -1.
Solving the system of equations by rearranging and using elimination yields the solution x = 5 and y = 3. Substituting these values into the target expression x - 2y gives 5 - 2(3), which evaluates to 5 - 6 = -1.

Step-by-Step Solution

1
Rearrange the first equation into standard form to align the variables.
5x−4y=135x - 4y = 13
Aligning the variables vertically makes it much easier to use the elimination method.
2
Multiply the entire second equation by 4.
8x+4y=528x + 4y = 52
This creates opposite coefficients (-4y and +4y) so the y variable will cancel out when the equations are added.
3
Add the rearranged first equation and the modified second equation together.
(5x−4y)+(8x+4y)=13+52⇒13x=65⇒x=5(5x - 4y) + (8x + 4y) = 13 + 52 \Rightarrow 13x = 65 \Rightarrow x = 5
Eliminating y isolates x, allowing us to find its numerical value.
4
Substitute x = 5 back into the original second equation to solve for y.
2(5)+y=13⇒10+y=13⇒y=32(5) + y = 13 \Rightarrow 10 + y = 13 \Rightarrow y = 3
This completes the solution pair (x, y) for the system of equations.
5
Substitute x = 5 and y = 3 into the requested expression x−2yx - 2y.
5−2(3)=5−6=−15 - 2(3) = 5 - 6 = -1
The question asks for the evaluated value of this specific expression, not just the solution coordinates.

Key Concept

Solving Systems of Linear Equations using elimination or substitution, followed by algebraic evaluation.
Question 57Question →

Given the system of equations below, what is the value of xx?

3x+2y=163x + 2y = 16
x−2y=0x - 2y = 0
Show answer & explanation

Answer: 4; x=4; x = 4; four

Answer

4
Because the coefficients of yy in the two equations are opposites (+2+2 and −2-2), the fastest approach is the elimination method. Adding the equations top-to-bottom yields (3x+x)+(2y−2y)=16+0(3x + x) + (2y - 2y) = 16 + 0, which simplifies directly to 4x=164x = 16. Dividing both sides by 44 gives x=4x = 4.

Step-by-Step Solution

1
Add the two equations together.
4x=164x = 16
The 2y2y and −2y-2y terms are opposites, so adding the equations eliminates the yy variable entirely.
2
Divide both sides of the resulting equation by 4.
x=4x = 4
This isolates the variable xx to find its exact numerical value.

Key Concept

Solving systems of linear equations using the elimination method.
Question 58Question →

To determine the critical points of a trajectory, a physicist needs to fully factor the quadratic expression 4x2−8x−604x^2 - 8x - 60. What is the completely factored equivalent of this expression?

Show answer & explanation

Answer: 4(x−5)(x+3)4(x - 5)(x + 3)

Answer

The completely factored form of the expression is 4(x−5)(x+3)4(x - 5)(x + 3).
The correct answer is found by first extracting the Greatest Common Factor (GCF), which is 44, leaving 4(x2−2x−15)4(x^2 - 2x - 15). Then, the inner trinomial x2−2x−15x^2 - 2x - 15 is factored into (x−5)(x+3)(x - 5)(x + 3) because −5-5 and 33 multiply to −15-15 and add to −2-2. Combining these gives the fully factored form 4(x−5)(x+3)4(x - 5)(x + 3).

Step-by-Step Solution

1
Identify and factor out the Greatest Common Factor (GCF) from all terms in the expression 4x2−8x−604x^2 - 8x - 60.
The GCF of 44, −8-8, and −60-60 is 44. Factoring it out yields 4(x2−2x−15)4(x^2 - 2x - 15).
Extracting the GCF first simplifies the remaining polynomial, making the trinomial easier to factor.
2
Factor the remaining quadratic trinomial x2−2x−15x^2 - 2x - 15 by finding two numbers that multiply to the constant term (−15-15) and add to the middle term's coefficient (−2-2).
The numbers are −5-5 and 33, so the trinomial factors to (x−5)(x+3)(x - 5)(x + 3).
Expanding (x−5)(x+3)(x - 5)(x + 3) gives x2+3x−5x−15x^2 + 3x - 5x - 15, which simplifies back to x2−2x−15x^2 - 2x - 15.
3
Combine the GCF and the factored binomials to write the completely factored form.
4(x−5)(x+3)4(x - 5)(x + 3)
This represents the original expression broken down into its prime polynomial factors.

Key Concept

Factoring polynomials by first extracting the Greatest Common Factor (GCF) and then factoring the resulting quadratic trinomial.
Estimated Time:1m 0s
Question 59Question →

Which ordered pair (x,y)(x, y) is the solution to the following system of linear equations?

3x+y=173x + y = 17
x−y=3x - y = 3
Show answer & explanation

Answer: (5,2)(5, 2)

Answer

The correct solution is the ordered pair (5,2)(5, 2).
The correct answer is (5, 2). By adding the two equations, the y terms cancel each other out, leaving 4x = 20. Dividing both sides by 4 gives x = 5. Substituting this value back into the second equation (5 - y = 3) allows us to solve for y, which is 2.

Step-by-Step Solution

1
Add the two equations together to eliminate the y variable.
(3x+y)+(x−y)=17+3⇒4x=20(3x + y) + (x - y) = 17 + 3 \Rightarrow 4x = 20
Since the coefficients of y are opposites (1 and -1), adding the equations cancels out the y terms.
2
Solve for x by dividing both sides of the resulting equation by 4.
x=5x = 5
Dividing by the coefficient of x isolates the variable.
3
Substitute the value of x back into the second original equation.
5−y=35 - y = 3
Substituting the known x value leaves an equation with only one variable, y.
4
Solve for y.
−y=3−5⇒−y=−2⇒y=2-y = 3 - 5 \Rightarrow -y = -2 \Rightarrow y = 2
Isolating y provides the final part of the solution.

Key Concept

Solving systems of linear equations using the elimination method
Question 60Question →

Consider the following system of linear equations:

4(x−2)+y=3x 4(x - 2) + y = 3x
2x−3(y+1)=−2 2x - 3(y + 1) = -2

What is the value of the expression x−yx - y?

Show answer & explanation

Answer: 2

Answer

2
The correct answer is derived by first simplifying both equations into standard form (x + y = 8 and 2x - 3y = 1). By applying the elimination method, we find the system's solution is x = 5 and y = 3. Finally, evaluating the specific requested expression gives us x - y = 5 - 3 = 2.

Step-by-Step Solution

1
Simplify the first equation.
x + y = 8
Equations must be in standard form to easily apply elimination or substitution.
2
Simplify the second equation.
2x - 3y = 1
Expanding the terms and grouping variables on one side creates the standard form.
3
Multiply the simplified first equation by 3.
3x + 3y = 24
This aligns the y-coefficients (3y and -3y) for the elimination method.
4
Add the modified first equation to the simplified second equation.
5x = 25
Adding the equations eliminates the y variable, allowing us to solve for x.
5
Divide both sides by 5.
x = 5
Isolates the variable x.
6
Substitute x = 5 into the simplified first equation (x + y = 8).
y = 3
This finds the value of the remaining variable, y.
7
Evaluate the target expression (x - y).
5 - 3 = 2
The question specifically asks for the difference between x and y, not just their individual values.

Key Concept

Solving systems of linear equations involving pre-simplification
PreviousPage 3 / 15Next
Algebra Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 3 | Examkin