Multiplying Polynomials
27 questions
Question 1Question →
Consider the following algebraic expression:
(2x−5y)2(x+2y)
If you were to fully expand this expression and combine all like terms, what would be the numerical coefficient of the x2y term?
Fill in the blanks below
The numerical coefficient of the x2y term is .
Show answer & explanation
Answer
-12
The correct answer is derived by first expanding the squared binomial: (2x−5y)2=4x2−20xy+25y2. Next, multiply this resulting trinomial by the remaining binomial: (4x2−20xy+25y2)(x+2y). To find the coefficient of the x2y term without expanding the entire expression, identify the term pairs whose product yields x2y. These pairs are (4x2)(2y)=8x2y and (−20xy)(x)=−20x2y. Combining these like terms gives 8x2y−20x2y=−12x2y. Thus, the numerical coefficient of the x2y term is −12.
Step-by-Step Solution
1
Expand the squared binomial (2x−5y)2.
(2x)2−2(2x)(5y)+(5y)2=4x2−20xy+25y2
Order of operations requires evaluating exponents before multiplying by the binomial (x+2y).
2
Set up the multiplication of the resulting trinomial and the remaining binomial.
(4x2−20xy+25y2)(x+2y)
Substitute the expanded form back into the original expression to continue the multiplication process.
3
Identify only the specific multiplications that will produce an x2y term.
The term 4x2 multiplied by 2y, and the term −20xy multiplied by x.
To find the coefficient of a specific term, it is most efficient to calculate only the products that result in that exact variable combination.
4
Calculate and combine the identified products to find the final coefficient.
(4x2)(2y)+(−20xy)(x)=8x2y−20x2y=−12x2y. The numerical coefficient is −12.
Combining these like terms gives the total x2y term in the fully expanded polynomial.
Key Concept
Multiplying Polynomials
Question 2Question →
A civil engineer is designing a rectangular water reservoir. The length of the reservoir is represented by the polynomial (4x−3) meters, and the cross-sectional area of the side wall is represented by (2x2+5x−1) square meters. If the total volume of the reservoir is calculated by multiplying its length by its cross-sectional area, which of the following polynomials represents the total volume in cubic meters?
8x3+14x2−19x+3
8x3+14x2−19x−3
8x3+20x2−4x−3
8x3+16x2−21x+3
Show answer & explanation
Answer: 8x3+14x2−19x+3
Answer
The correct polynomial representing the volume is 8x3+14x2−19x+3.
The correct expression is found by properly distributing both terms of the binomial (4x−3) to all three terms of the trinomial (2x2+5x−1), maintaining correct signs, and accurately combining the resulting like terms.
Step-by-Step Solution
1
Set up the multiplication of the two polynomials.
V=(4x−3)(2x2+5x−1)
The problem states that the volume is calculated by multiplying the length by the cross-sectional area.
2
Distribute the first term of the binomial, 4x, to every term in the trinomial.
4x(2x2)+4x(5x)+4x(−1)=8x3+20x2−4x
The distributive property requires multiplying every term in the second polynomial by the first term of the first polynomial.
3
Distribute the second term of the binomial, −3, to every term in the trinomial.
−3(2x2)−3(5x)−3(−1)=−6x2−15x+3
Carefully apply the negative sign to all terms during multiplication, noting that a negative times a negative yields a positive.
4
Combine the results and group the like terms.
8x3+(20x2−6x2)+(−4x−15x)+3
Like terms (terms with the exact same variable configuration) must be grouped together to simplify the expression.
5
Perform the final addition and subtraction to simplify completely.
8x3+14x2−19x+3
This yields the fully simplified polynomial in standard form.
Key Concept
Multiplying a binomial by a trinomial using the Distributive Property
Question 3Question →
Simplify the following polynomial expression completely:
(3x−2y)(2x2−xy+4y2)−x(x−3y)2
Which of the following represents the fully expanded and simplified result?
5x3−x2y+5xy2−8y3
5x3−13x2y+23xy2−8y3
5x3−7x2y+23xy2−8y3
7x3−13x2y+23xy2−8y3
Show answer & explanation
Answer: 5x3−x2y+5xy2−8y3
Answer
The completely simplified expression is 5x3−x2y+5xy2−8y3.
The correct answer is derived by carefully following the order of operations: first expanding both the left (3x−2y)(2x2−xy+4y2) and right x(x−3y)2 sides of the subtraction independently, and then combining the resulting polynomials. Distributing the negative sign correctly during the final subtraction yields 5x3−x2y+5xy2−8y3.
Step-by-Step Solution
1
Expand the product of the binomial and the trinomial: (3x−2y)(2x2−xy+4y2).
6x3−3x2y+12xy2−4x2y+2xy2−8y3
Every term in the first binomial must be multiplied by every term in the trinomial to ensure full expansion.
2
Combine the like terms in the first expanded expression.
6x3−7x2y+14xy2−8y3
Combining like terms simplifies the polynomial before performing the final subtraction operation.
3
Expand the squared binomial in the second part of the expression: (x−3y)2.
x2−6xy+9y2
A perfect square binomial (a−b)2 expands to a2−2ab+b2.
4
Distribute the x into the expanded squared binomial: x(x2−6xy+9y2).
x3−6x2y+9xy2
The outer monomial must be distributed to each term in the trinomial.
5
Subtract the second result from the first result: (6x3−7x2y+14xy2−8y3)−(x3−6x2y+9xy2).
6x3−7x2y+14xy2−8y3−x3+6x2y−9xy2
Subtracting a polynomial requires distributing the negative sign to every term inside the parentheses.
6
Combine all remaining like terms.
5x3−x2y+5xy2−8y3
Grouping x3, x2y, and xy2 terms produces the final simplified answer in standard form.
Key Concept
Multiplying complex polynomials and applying subtraction across multiple terms
Question 4Question →
When the polynomial expression (3x−2y)(Ax2+4xy−5y2) is fully expanded and simplified, the coefficient of the x2y term is −4. What is the value of the constant A?
2
8
−4
−8
Show answer & explanation
Answer: 8
Answer
The value of the constant A is 8.
To find the coefficient of the x2y term, we do not need to expand the entire expression. We only need to find the specific multiplications that result in an x2y term. Multiplying the x term from the first polynomial by the xy term from the second gives 3x×4xy=12x2y. Multiplying the y term from the first polynomial by the x2 term from the second gives −2y×Ax2=−2Ax2y. Combining these gives a total x2y coefficient of (12−2A). Since the problem states this coefficient is −4, we set up the equation 12−2A=−4. Subtracting 12 from both sides gives −2A=−16. Dividing by −2 gives A=8.
Step-by-Step Solution
1
Identify the products that result in an x2y term when expanding the polynomials.
The relevant terms are (3x)(4xy) and (−2y)(Ax2).
Only these specific combinations of terms will produce the exact variable configuration x2y.
2
Multiply these specific terms to find their coefficients.
(3x)(4xy)=12x2y and (−2y)(Ax2)=−2Ax2y.
This isolates the parts of the expansion that contribute to the final x2y coefficient.
3
Combine the coefficients and set them equal to the given total coefficient of −4.
12−2A=−4
The algebraic sum of the individual x2y coefficients must equal the final simplified coefficient provided in the problem.
4
Solve the linear equation for A.
−2A=−4−12⇒−2A=−16⇒A=8
Standard algebraic isolation of the variable A gives the final answer.
Key Concept
Identifying specific terms in polynomial multiplication and solving for an unknown coefficient.
Estimated Time:1m 30s
Question 5Question →
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). What is the numerical coefficient of the a2b term?
Show answer & explanation
Answer: -39
Answer
-39
To find the coefficient of the a2b term, first expand the squared binomial (3a−4b)2 to get the trinomial 9a2−24ab+16b2. Next, multiply this trinomial by (2a+b). The specific terms that result in a2b are the product of 9a2 and b (which yields 9a2b) and the product of −24ab and 2a (which yields −48a2b). Combining these like terms gives 9a2b−48a2b=−39a2b. Therefore, the numerical coefficient is -39.
Step-by-Step Solution
1
Expand the squared binomial (3a−4b)2.
9a2−24ab+16b2
Order of operations requires resolving the exponent before multiplying by the binomial (2a+b).
2
Set up the multiplication of the resulting trinomial and the binomial.
(9a2−24ab+16b2)(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 a2b term.
(9a2)(b)=9a2b and (−24ab)(2a)=−48a2b
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=−39a2b
Combining these terms gives the final, simplified coefficient for the a2b term.
Key Concept
Multiplying polynomials, expanding binomial squares, and combining like terms.
Question 6Question →
Which of the following represents the fully expanded and simplified form of the polynomial expression (2x−3)(x2−4x+5)?
2x3−11x2+22x−15
2x3−5x2−2x−15
2x3−15
2x3+11x−15
Show answer & explanation
Answer: 2x3−11x2+22x−15
Answer
The simplified polynomial is 2x3−11x2+22x−15.
The correct answer is found by distributing each term in the binomial to each term in the trinomial and combining like terms. Distributing 2x yields 2x3−8x2+10x, and distributing −3 yields −3x2+12x−15. Combining −8x2 with −3x2 gives −11x2, and combining 10x with 12x gives 22x, resulting in the final simplified expression 2x3−11x2+22x−15.
Step-by-Step Solution
1
Distribute the first term of the binomial, 2x, to each term in the trinomial (x2−4x+5).
2x(x2)+2x(−4x)+2x(5)=2x3−8x2+10x
To multiply polynomials, every term in the first polynomial must be multiplied by every term in the second.
2
Distribute the second term of the binomial, −3, to each term in the trinomial.
−3(x2)−3(−4x)−3(5)=−3x2+12x−15
It is critical to carry the negative sign when distributing to ensure the signs of the resulting terms are correct.
3
Add the results from both distribution steps and group the like terms together.
2x3+(−8x2−3x2)+(10x+12x)−15
Grouping terms with the same variable and exponent makes it easier to simplify.
4
Combine the coefficients of the grouped like terms.
2x3−11x2+22x−15
Combining like terms provides the expression in its fully simplified, standard polynomial form.
Key Concept
Multiplying Polynomials
Question 7Question →
Expand the given product of a binomial and a trinomial, combine like terms, and determine the values of the missing coefficients.
Fill in the blanks below
When the expression (3x−5)(2x2+4x−7) is fully expanded and simplified, the resulting polynomial in standard form is 6x3+ x2+ x+35.
Show answer & explanation
Answer
The coefficient for the x^2 term is 2, and the coefficient for the x term is -41.
By distributing each term of the binomial (3x−5) to every term of the trinomial (2x2+4x−7), we get six terms. Grouping and combining the like terms yields an x2 coefficient of 12−10=2 and an x coefficient of −21−20=−41.
Step-by-Step Solution
1
Distribute the first term of the binomial, 3x, to each term in the trinomial.
3x(2x2)+3x(4x)+3x(−7)=6x3+12x2−21x
To apply the distributive property for the first part of the polynomial multiplication.
2
Distribute the second term of the binomial, −5, to each term in the trinomial.
−5(2x2)−5(4x)−5(−7)=−10x2−20x+35
To apply the distributive property for the second part, being careful to carry the negative sign.
3
Combine the results and group the like terms.
6x3+(12x2−10x2)+(−21x−20x)+35
To arrange the expression so that coefficients of terms with the same exponent can be simplified.
4
Simplify the grouped terms.
6x3+2x2−41x+35
By adding the coefficients: 12−10=2 and −21−20=−41.
Key Concept
Distributive Property with Polynomials
Alternative Method
You can use the 'box method' (a grid) by creating a 2x3 table with 3x and −5 on the rows, and 2x2, 4x, and −7 on the columns. Multiply the row and column headers to fill each cell, then add all the cells together along the diagonals to combine like terms.
Estimated Time:1m 30s
Question 8Question →
Which of the following represents the fully simplified form of the expression (3x−4)2−x(2x+5)?
7x2−19x+16
7x2−5x+16
7x2−29x−16
7x2−29x+16
Show answer & explanation
Answer: 7x2−29x+16
Answer
The correct simplified expression is 7x2−29x+16.
Expanding the squared binomial yields 9x2−24x+16. Distributing the −x yields −2x2−5x. Combining the like terms gives (9x2−2x2)+(−24x−5x)+16, which simplifies perfectly to 7x2−29x+16.
Step-by-Step Solution
1
Expand the squared binomial (3x−4)2.
(3x)2−2(3x)(4)+42=9x2−24x+16
Using the perfect square formula (a−b)2=a2−2ab+b2 is required to properly multiply the binomial by itself.
2
Distribute the term −x across the binomial (2x+5).
−x(2x)−x(5)=−2x2−5x
The distributive property must be applied, ensuring the negative sign is carried to both terms inside the parentheses.
3
Combine the results from the previous steps and combine like terms.
(9x2−24x+16)+(−2x2−5x)=(9x2−2x2)+(−24x−5x)+16=7x2−29x+16
Like terms (those with the same variable raised to the same power) must be grouped and added together to find the fully simplified polynomial form.
Key Concept
Expanding perfect square binomials and distributing monomial terms correctly.
Question 9Question →
Multiply the binomial (3x−4) by the trinomial (5x2−2x−7). After fully expanding the product and combining all like terms, what is the numerical coefficient of the x term?
Show answer & explanation
Answer: -13
Answer
The coefficient of the x term is -13.
The coefficient of the x term is found by locating all the x terms in the expanded expression (−21x and 8x) and adding their coefficients together (−21+8=−13).
Step-by-Step Solution
1
Distribute the term 3x to every term in the trinomial (5x2−2x−7).
3x(5x2)+3x(−2x)+3x(−7)=15x3−6x2−21x
The distributive property requires multiplying each term of the first polynomial by each term of the second polynomial.
2
Distribute the term −4 to every term in the trinomial (5x2−2x−7).
−4(5x2)−4(−2x)−4(−7)=−20x2+8x+28
This completes the distribution for the second term of the binomial, taking care to properly multiply the negative signs.
3
Write out the fully expanded polynomial before combining like terms.
15x3−6x2−21x−20x2+8x+28
Listing all terms together makes it easier to identify which terms share the same variables and exponents.
4
Identify and combine the like terms that contain exactly the variable x to the first power.
−21x+8x=−13x
The question specifically asks for the coefficient of the x term, so we combine −21x and 8x.
Key Concept
Multiplying polynomials and combining like terms
Question 10Question →
What is the product of the binomials (2x−1) and (x+4)?
2x2−4
2x2−7x−4
2x2+7x−4
9x3−4
Show answer & explanation
Answer: 2x2+7x−4
Answer
The correct expression is 2x2+7x−4.
By applying the FOIL method, we multiply the First terms to get 2x2, the Outer terms to get 8x, the Inner terms to get −x, and the Last terms to get −4. Combining the like terms 8x and −x results in the simplified polynomial 2x2+7x−4.
Step-by-Step Solution
1
Multiply the First terms.
2x×x=2x2
This is the 'F' step in the FOIL method.
2
Multiply the Outer terms.
2x×4=8x
This is the 'O' step in the FOIL method.
3
Multiply the Inner terms.
−1×x=−x
This is the 'I' step in the FOIL method.
4
Multiply the Last terms.
−1×4=−4
This is the 'L' step in the FOIL method.
5
Combine the resulting terms.
2x2+8x−x−4=2x2+7x−4
Combine the like middle terms (8x and −x) to simplify the final expression.
Key Concept
Multiplying binomials using the distributive property (FOIL method) and combining like terms.
Question 11Question →
Multiply the binomials (x−5) and (x+2). Fill in the missing values for the resulting expanded expression x2+bx+c.
Fill in the blanks below
The value of b is .
The value of c is .
The value of c is .
Show answer & explanation
Answer
The value of b is -3 and the value of c is -10.
By applying the FOIL method, you multiply the First terms (x×x=x2), Outer terms (x×2=2x), Inner terms (−5×x=−5x), and Last terms (−5×2=−10). Combining the like terms (2x−5x) results in −3x. The fully expanded expression is x2−3x−10. Comparing this to the standard form x2+bx+c, we find that the coefficient for the middle term b is −3 and the constant term c is −10.
Step-by-Step Solution
1
Use the FOIL method to distribute the terms in the binomials.
(x×x)+(x×2)+(−5×x)+(−5×2)
Each term in the first binomial, including its sign, must be multiplied by each term in the second binomial.
2
Perform the multiplications for each pair.
x2+2x−5x−10
This expands the product using the rules for multiplying positive and negative numbers.
3
Combine the like terms.
x2−3x−10
Adding the middle terms (2x and −5x) simplifies the expression into the standard form x2+bx+c.
Key Concept
Multiplying Polynomials
Question 12Question →
What is the expanded form of the product (a−3)(a+7)?
a2−21
a2+4a−21
a2−4a−21
a2−17a
Show answer & explanation
Answer: a2+4a−21
Answer
a2+4a−21
The expression is properly expanded by distributing each term in the first binomial to each term in the second binomial, resulting in a2+7a−3a−21. Combining the like terms (7a−3a) yields the final simplified polynomial a2+4a−21.
Step-by-Step Solution
1
Multiply the first terms in each binomial.
a×a=a2
To begin distributing the first term of the first binomial.
2
Multiply the outer terms.
a×7=7a
To complete the distribution of the first term.
3
Multiply the inner terms.
−3×a=−3a
To begin distributing the second term of the first binomial.
4
Multiply the last terms in each binomial.
−3×7=−21
To complete the distribution of the second term.
5
Combine the like middle terms to simplify the expression.
7a+(−3a)=4a
To write the polynomial in its fully simplified standard form, resulting in a2+4a−21.
Key Concept
Multiplying Binomials
Question 13Question →
What is the coefficient of the k term in the expanded product of (3k−2) and (k+5)?
Show answer & explanation
Answer: 13
Answer
13
When expanding (3k−2)(k+5), the k terms are generated by multiplying the outer terms (3k×5=15k) and the inner terms (−2×k=−2k). Combining these gives 15k−2k=13k. Therefore, the coefficient of the k term is 13.
Step-by-Step Solution
1
Identify the terms that will produce a k term when multiplied.
The outer terms are 3k and 5. The inner terms are −2 and k.
Only the product of a variable term and a constant term will result in a term with degree 1 (k).
2
Multiply the outer terms.
(3k)(5)=15k
This is the 'Outer' multiplication step in expanding two binomials.
3
Multiply the inner terms.
(−2)(k)=−2k
This is the 'Inner' multiplication step in expanding two binomials.
4
Combine the resulting like terms.
15k+(−2k)=13k
Combining like terms gives the final k term of the expanded polynomial.
5
Identify the coefficient.
13
The coefficient is the numerical factor of the k term.
Key Concept
Multiplying polynomials and combining like terms
Question 14Question →
Which of the following is equivalent to the expression (4p−1)(p+2)?
4p2+7p−2
4p2−2
4p2+9p−2
11p−2
Show answer & explanation
Answer: 4p2+7p−2
Answer
The equivalent expression is 4p2+7p−2.
Using the distributive property (often remembered as FOIL), each term in the first binomial must be multiplied by each term in the second binomial. The products are 4p2, 8p, −p, and −2. Combining the like terms (8p and −p) yields 7p, making the final simplified expression 4p2+7p−2.
Step-by-Step Solution
1
Multiply the first terms of each binomial.
4p×p=4p2
This establishes the quadratic term of the final polynomial.
2
Multiply the outer terms.
4p×2=8p
This finds the first part of the linear middle term.
3
Multiply the inner terms.
−1×p=−p
This finds the second part of the linear middle term.
4
Multiply the last terms.
−1×2=−2
This finds the constant term of the final polynomial.
5
Combine the like terms.
4p2+(8p−p)−2=4p2+7p−2
Simplifying the expression converts it into standard polynomial form.
Key Concept
Multiplying Binomials
Question 15Question →
Determine the missing coefficient in the area expression.
Fill in the blanks below
A rectangle has a length of (3x−4) units and a width of (x+6) units. The area of the rectangle can be represented by the expanded polynomial $3x^2 + x - 24$ square units.
Show answer & explanation
Answer
14
The area of the rectangle is calculated by multiplying its length and width: (3x−4)(x+6). Distributing the terms, the outer product is 18x and the inner product is −4x. Combining these like terms (18x−4x) results in 14x. Therefore, the missing coefficient for the x term is 14.
Step-by-Step Solution
1
Set up the multiplication for the area of the rectangle.
(3x−4)(x+6)
The area of a rectangle is found by multiplying its length and its width.
2
Multiply the binomials using the distributive property.
3x2+18x−4x−24
Each term in the first binomial must be multiplied by each term in the second binomial.
3
Combine the like terms containing the x variable.
18x−4x=14x
Simplifying the expression requires combining terms that share the same variable and exponent.
Key Concept
Multiplying binomials and combining like terms
Question 16Question →
Calculate the product of the binomials (m−6) and (m+2). What is the resulting polynomial in simplest form?
m2−4m−12
m2−12
m2+4m−12
m2−4m+12
Show answer & explanation
Answer: m2−4m−12
Answer
The correct answer is the polynomial m2−4m−12.
The correct answer is obtained by distributing each term in the first binomial to each term in the second binomial, which gives m2+2m−6m−12. Combining the like terms (2m−6m) yields −4m, resulting in the final simplified polynomial m2−4m−12.
Step-by-Step Solution
1
Apply the distributive property to multiply each term in the first binomial by each term in the second binomial.
m⋅m+m⋅2−6⋅m−6⋅2
To expand the product of the two binomial expressions.
2
Perform the individual multiplications for each term.
m2+2m−6m−12
To simplify the expression before combining any like terms.
3
Combine the like terms (the middle terms containing m).
m2−4m−12
To write the polynomial in its standard, fully simplified form.
Key Concept
Multiplying Polynomials
Alternative Method
You can use the area model (or box method) to organize the multiplication. Draw a 2x2 grid, place the terms m and −6 along the top edge, and m and 2 along the side. Multiply the intersecting terms to fill the four cells (m2, −6m, 2m, and −12), and then sum all the cells to find the final polynomial.
Estimated Time:45s
Question 17Question →
The expression (2y−7)2 is expanded into the standard polynomial form ay2+by+c. What is the value of the coefficient b?
Show answer & explanation
Answer: -28
Answer
-28
Expanding the perfect square binomial (2y−7)2 requires multiplying (2y−7) by itself. Using the distributive property, the outer product is (2y)(−7)=−14y and the inner product is (−7)(2y)=−14y. Combining these like terms gives a total middle term of −28y. Because the required polynomial form is ay2+by+c, the value of the coefficient b is exactly -28.
Step-by-Step Solution
1
Rewrite the expression as a product of two binomials.
(2y−7)(2y−7)
Writing the binomial twice makes it easier to correctly apply the distributive property (FOIL method) and avoid missing the middle terms.
2
Distribute the terms.
4y2−14y−14y+49
Multiplying the First terms, Outer terms, Inner terms, and Last terms ensures that every term in the first binomial is multiplied by every term in the second binomial.
3
Combine the like terms.
4y2−28y+49
The middle terms, −14y and −14y, have the same variable raised to the same power, so their coefficients can be added together.
4
Identify the target coefficient.
−28
The standard polynomial form is given as ay2+by+c. The value of b corresponds to the coefficient of the y term.
Key Concept
Expanding Perfect Square Binomials
Question 18Question →
When the algebraic expression (4x−3)(2x2+5x−1) is fully expanded and simplified into standard polynomial form, what is the value of the coefficient of the x term?
Show answer & explanation
Answer: -19
Answer
-19
The correct answer is −19 because the x term is exclusively formed by the sum of the products (4x)(−1) and (−3)(5x), which equals −4x−15x=−19x.
Step-by-Step Solution
1
Identify the goal.
We need to find the coefficient of the x term in the product of (4x−3) and (2x2+5x−1).
Focusing only on the terms that produce an x term saves time compared to expanding the entire polynomial expression.
2
Find the first partial product that results in an x term.
Multiply the x term from the binomial (4x) by the constant term from the trinomial (−1) to get −4x.
An x term multiplied by a constant results in an x term.
3
Find the second partial product that results in an x term.
Multiply the constant term from the binomial (−3) by the x term from the trinomial (5x) to get −15x.
A constant multiplied by an x term results in an x term.
4
Combine the partial products.
−4x+(−15x)=−19x
To find the total x term in the expanded polynomial, all like terms must be combined.
5
Identify the final coefficient.
The coefficient of −19x is −19.
The coefficient is the numerical multiplier of the variable.
Key Concept
Multiplying Polynomials and Combining Like Terms
Question 19Question →
When the expression (3x−4)(2x+5)−x(x−2) is fully expanded and simplified into standard polynomial form, what is the value of the coefficient of x?
Show answer & explanation
Answer: 9
Answer
9
Expanding the binomials gives 6x2+7x−20. Distributing −x gives −x2+2x. Combining like terms results in 5x2+9x−20. The question asks for the coefficient of x, which is the number multiplying the x term: 9.
Step-by-Step Solution
1
Multiply the binomials (3x−4) and (2x+5).
6x2+15x−8x−20, which simplifies to 6x2+7x−20.
To expand the first part of the expression using the distributive property (FOIL).
2
Distribute −x across the binomial (x−2).
−x2+2x.
To expand the second part of the expression, ensuring the negative sign is applied to both terms inside the parentheses.
3
Combine the expanded parts.
(6x2+7x−20)+(−x2+2x)=5x2+9x−20.
To simplify the full expression into standard polynomial form by grouping like terms.
4
Identify the coefficient of x.
The coefficient of x in 5x2+9x−20 is 9.
This answers the specific question asked.
Key Concept
Multiplying polynomials and combining like terms
Question 20Question →
The dimensions of a rectangular box are given by the expressions 5x, (2x−3), and (x+4). The volume of the box is the product of its three dimensions. When the volume is expressed as a fully simplified polynomial in the form ax3+bx2+cx, determine the value of the coefficient b.
Fill in the blanks below
The value of b is .
Show answer & explanation
Answer
25
To find the coefficient b, you must expand the product of the three dimensions. First, multiply the binomials using FOIL: (2x−3)(x+4)=2x2+8x−3x−12=2x2+5x−12. Then, distribute 5x across the resulting trinomial: 5x(2x2+5x−12)=10x3+25x2−60x. Comparing this to the specified polynomial form ax3+bx2+cx, the term with x2 is 25x2, making the coefficient b=25.
Step-by-Step Solution
1
Set up the volume expression by multiplying the three given dimensions.
V=5x(2x−3)(x+4)
The volume of a rectangular box is calculated as length × width × height.
2
Multiply the two binomials using the FOIL method.
(2x−3)(x+4)=2x2+8x−3x−12=2x2+5x−12
It is mathematically sound and often easier to multiply the binomials first before distributing the monomial.
3
Distribute the monomial 5x to each term in the resulting trinomial.
5x(2x2+5x−12)=10x3+25x2−60x
To fully expand the polynomial, every term inside the parentheses must be multiplied by 5x.
4
Identify the value of b by matching the result to the given standard form ax3+bx2+cx.
b=25
The variable b represents the coefficient of the x2 term in the fully expanded polynomial.
Key Concept
Multiplying Polynomials and the Distributive Property
Page 1 / 2Next