Question

write an equivalent expression using the fewest number of terms.

1. $(4x^{4}+x^{3}+2x^{2})$ and $(3x^{3}+x^{2}+4x)$
2. $(ab^{2}+13b - 4a)$ and $(3ab^{2}+a + 7b)$
3. $(n^{3}+12n^{2}-2n)$ and $(-n^{4}+3n^{3}-4n^{2})$

model the situation with the sum of polynomials. then simplify the sum.

1. the width of a rectangle is represented by $4x$, and its length is represented by $(3x + 2)$. write a polynomial for the perimeter of the rectangle.

Explanation:

Step1: Combine like - terms for 1

For the polynomials $(4x^{4}+x^{3}+2x^{2})$ and $(3x^{3}+x^{2}+4x)$, combine terms with the same power of $x$.
$4x^{4}+(x^{3}+3x^{3})+(2x^{2}+x^{2}) + 4x=4x^{4}+4x^{3}+3x^{2}+4x$

Step2: Combine like - terms for 2

For the polynomials $(ab^{2}+13b - 4a)$ and $(3ab^{2}+a + 7b)$, combine like - terms.
$(ab^{2}+3ab^{2})+(13b + 7b)+(-4a + a)=4ab^{2}+20b-3a$

Step3: Combine like - terms for 3

For the polynomials $(n^{3}+12n^{2}-2n)$ and $(-n^{4}+3n^{3}-4n^{2})$, combine like - terms.
$-n^{4}+(n^{3}+3n^{3})+(12n^{2}-4n^{2})-2n=-n^{4}+4n^{3}+8n^{2}-2n$

Step4: Find perimeter for 4

The perimeter $P$ of a rectangle is $P = 2(l + w)$, where $l$ is the length and $w$ is the width. Given $l=3x + 2$ and $w = 4x$.
$P=2((3x + 2)+4x)=2(3x+4x + 2)=2(7x + 2)=14x+4$

Answer:

1. $4x^{4}+4x^{3}+3x^{2}+4x$
2. $4ab^{2}+20b-3a$
3. $-n^{4}+4n^{3}+8n^{2}-2n$
4. $14x + 4$