Answers Algebra 2 MP 234 CHSMultiplication of Polynomials

Multiplication of Polynomials Answers

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The dimensions of the base of Box 1 are:width: x; length: 3width: x; length: x + 3width: x; length: x – 3width: x; length: 3x

width: x; length: 3

width: x; length: x + 3

width: x; length: x – 3

width: x; length: 3x

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3x3x23x34x

3x2

3x3

Box 1: Dimensions: x by 3x by x3Area of base = x(3x) = 3x2Box 2: Dimensions: x by 4x – 1 by x3Area of base = x(4x – 1) = 4x2 – xComplete the statements about the number of terms in the polynomial representing the volume of each box. Box 1’s volume will be a . Box 2’s volume will be a . Explain your reasoning.

Answer:

monomial

Dimensions of Box 1: x by 3x by x3 The volume of Box 1 is given by:

3x7

3x6

3x5

5x3

Why does this make sense?

You use the power of a product law of exponents to combine the exponents; this law says to add the exponents.

You use the power of a power law of exponents to combine the exponents; this law says to multiply the exponents.

You use the product of powers law of exponents to combine the exponents; this law says to add the exponents.

Box 2: Degree of length: Degree of width: Degree of height: Degree of volume:

Answer:

1 1 3 5

The productsumdifferencequotient of the degrees of each factor is the degree of the product.

product

sum

difference

quotient

Box 3: Length = 4 + xWidth = xHeight = x2 + 1The polynomial that represents the volume of Box 3 has a degree of .

Answer:

What is the volume of Box 3?

4x3 + 2x2 + 4x + 1

x4 + 4x3 + x2 + 4x

x4 + 4x3 + 2x2 + 4x

x4 + 2x3 + x2 + 4

What is the base area of Box 3?x2 + x

Answer:

1 4

The dimensions of the base of Box 1 are:

width: x; length: 3

width: x; length: x + 3

width: x; length: x – 3

width: x; length: 3x

3x3x23x34x

3x2

3x3

Dimensions of Box 1: x by 3x by x3 The volume of Box 1 is given by:

3x7

3x6

3x5

5x3

Suppose Celine wants to choose a box that maximizes the amount of cereal it can hold.Volume of Box 1: 3x5Volume of Box 2: 4x5 – x4If Celine decides the width of the cereal boxes will be greater than 1, which box will hold more cereal? Explain.

Answer:

Box 2 will hold more cereal. To compare the two volumes, subtract the volume of Box 1 from the volume of Box 2: $(4x^5 - x^4) - (3x^5) = x^5 - x^4$. Since Celine decided that the width, $x$, will be greater than 1, $x^5$ will always be greater than $x^4$. This makes the difference $(x^5 - x^4)$ a positive value, meaning Box 2's volume is greater than Box 1's volume for all $x > 1$.

Box 2: Degree of length: Degree of width: ⇒ 1 Degree of height: ⇒ 3 Degree of volume: ⇒ 5

Answer:

Why does this make sense?

You use the power of a product law of exponents to combine the exponents; this law says to add the exponents.

You use the power of a power law of exponents to combine the exponents; this law says to multiply the exponents.

You use the product of powers law of exponents to combine the exponents; this law says to add the exponents.

Box 3: Length = 4 + xWidth = xHeight = x2 + 1The polynomial that represents the volume of Box 3 has a degree of .

Answer:

What is the volume of Box 3?

4x3 + 2x2 + 4x + 1

x4 + 4x3 + x2 + 4x

x4 + 4x3 + 2x2 + 4x

x4 + 2x3 + x2 + 4

What is the base area of Box 3?x2 + x

Answer:

The dimensions of the base of Box 2 are:width: x; length: 3width: x; length: 4x – 1width: x; length: x – 4width: x; length: 4x + 1

width: x; length: 3

width: x; length: 4x – 1

width: x; length: x – 4

width: x; length: 4x + 1

4x2 – x 4x2 – 14x – 45x – 1

4x2 – x

4x2 – 1

4x – 4

5x – 1

Dimensions of Box 2: x by 4x – 1 by x3The volume of Box 2 is given by:

4x5 – x4

4x6 – x3

4x5 + x4

x3 + 5x –1

The dimensions of the base of Box 2 are:

width: x; length: 3

width: x; length: 4x – 1

width: x; length: x – 4

width: x; length: 4x + 1

4x2 – x 4x2 – 14x – 45x – 1

4x2 – x

4x2 – 1

4x – 4

5x – 1

Explain your reasoning.

Answer:

Box 1’s volume is modeled by a monomial times a monomial, so it will be a monomial. Box 2’s volume is modeled by a monomial times a binomial, so it will be a binomial.

Dimensions of Box 2: x by 4x – 1 by x3The volume of Box 2 is given by:

4x5 – x4

4x6 – x3

4x5 + x4

x3 + 5x –1

Which did you include in your response?

If a box has a greater volume, it can hold more cereal.

Box 2 can hold more cereal. Although the value of x is unknown, it represents a length, so it must be greater than zero. For any x > 1, the volume of Box 2 is greater than the volume of Box 1.

The product✔ sumdifferencequotient of the degrees of each factor is the degree of the product.

product

✔ sum

difference

quotient

Which did you include in your response?Box 1’s volume is modeled by a monomial times a monomial, so it will be a monomial.Box 2’s volume is modeled by a monomial times a binomial, so it will be a binomial.

Box 1’s volume is modeled by a monomial times a monomial, so it will be a monomial.

Box 2’s volume is modeled by a monomial times a binomial, so it will be a binomial.

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Multiplication of Polynomials Answers

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