Question

problems 3–4: fill in the blank with the polynomial that makes each equation true. write each polynomial as a sum using the fewest number of terms.

1. $(-4x^3 + 6x^4 - 9) + (\underline{quadquad}) = 8x^5 - 3x^4 + 7x^3 - 2x^2 + 4$
2. $(13x^4 - 5x^3 + x^2 + 10) - (\underline{quadquad}) = 15x^4 + 3x^3 - x^2 + 2x$
3. write $f(x)$ as a sum using the fewest number of terms. use the diagram if it helps with your thinking.

$f(x) = (x - 3)(x^2 - 4x)$
$f(x) = \underline{quadquad}$

1. write $h(x)$ as a sum using the fewest number of terms. use the diagram if it helps with your thinking.

$h(x) = (-2x^2 - 3x)(x^3 + 4x^2 - 5)$
$h(x) = \underline{quadquad}$

Explanation:

Problem 3

Step1: Isolate the unknown polynomial

Let the unknown polynomial be $P(x)$. Rearrange the equation:
$P(x) = 8x^5 - 3x^4 + 7x^3 - 2x^2 + 4 - (-4x^5 + 6x^3 - 9)$

Step2: Distribute the negative sign

$P(x) = 8x^5 - 3x^4 + 7x^3 - 2x^2 + 4 + 4x^5 - 6x^3 + 9$

Step3: Combine like terms

$P(x) = (8x^5+4x^5) + (-3x^4) + (7x^3-6x^3) + (-2x^2) + (4+9)$
$P(x) = 12x^5 - 3x^4 + x^3 - 2x^2 + 13$

Problem 4

Step1: Isolate the unknown polynomial

Let the unknown polynomial be $Q(x)$. Rearrange the equation:
$Q(x) = (13x^4 - 5x^3 + x^2 + 10) - (15x^4 + 3x^3 - x^2 + 2x)$

Step2: Distribute the negative sign

$Q(x) = 13x^4 - 5x^3 + x^2 + 10 - 15x^4 - 3x^3 + x^2 - 2x$

Step3: Combine like terms

$Q(x) = (13x^4-15x^4) + (-5x^3-3x^3) + (x^2+x^2) + (-2x) + 10$
$Q(x) = -2x^4 - 8x^3 + 2x^2 - 2x + 10$

Problem 5

Step1: Expand using distributive property

Multiply each term in $(x-3)$ by each term in $(x^2-4x)$:
$f(x) = x(x^2-4x) - 3(x^2-4x)$

Step2: Compute individual products

$f(x) = x^3 - 4x^2 - 3x^2 + 12x$

Step3: Combine like terms

$f(x) = x^3 + (-4x^2-3x^2) + 12x$
$f(x) = x^3 - 7x^2 + 12x$

Problem 6

Step1: Expand using distributive property

First multiply $(-2x^2-3x)$ by each term in $(x^3+4x^2-5)$:
$h(x) = -2x^2(x^3+4x^2-5) - 3x(x^3+4x^2-5)$

Step2: Compute individual products

$h(x) = -2x^5 - 8x^4 + 10x^2 - 3x^4 - 12x^3 + 15x$

Step3: Combine like terms

$h(x) = -2x^5 + (-8x^4-3x^4) - 12x^3 + 10x^2 + 15x$
$h(x) = -2x^5 - 11x^4 - 12x^3 + 10x^2 + 15x$

Answer:

1. $\boldsymbol{12x^5 - 3x^4 + x^3 - 2x^2 + 13}$
2. $\boldsymbol{-2x^4 - 8x^3 + 2x^2 - 2x + 10}$
3. $\boldsymbol{x^3 - 7x^2 + 12x}$
4. $\boldsymbol{-2x^5 - 11x^4 - 12x^3 + 10x^2 + 15x}$