Question

1. find each of the following polynomial sums. write your final answers in standard form.

(a) $(-5x^2 + 3x - 2)+(7x^2 + x + 8)$
(b) $(4x + x^2 - 9)+(11 - 3x + 5x^2)$
(c) $(x^3 - 2x + 1)+(-7x^3 + 3x^2 - 6x)$
(d) $(8x^3 - x^2 + 14x - 5)+(x - 8x^3 + 2x^2 + 2)$

1. find each of the following polynomial differences. write your final answers in standard form.

(a) $(3x^2 - x + 7)-(x^2 - 5x + 4)$
(b) $(-x^2 + 6x - 2)-(2x^2 + 9x - 7)$
(c) $(8x + 4x^2 - 5)-(-2x^3 + 4)$
(d) $(x^3 - 6x^2 + 2x + 11)-(9x + 13 - 4x^2 + 2x^3)$

can certainly have polynomial functions as well as expressions. if $f(x)=x^3 - 2x^2 + 8x - 1$ and $=7x^2 - 4x + 3$ then find an expression for each of the following in standard form:
$f(x)+g(x)$
(b) $f(x)-g(x)$

reasoning

1. polynomial expressions act a lot like integers because the structure of polynomials is based on the structure of integers. based on the statement below about integers, make a statement about polynomials.

statement about integers: an integer added to an integer gives an integer.
statement about polynomials:

Explanation:

Problem 6(a)

Step1: Group like terms

$(-5x^2 + 7x^2) + (3x + x) + (-2 + 8)$

Step2: Combine like terms

$2x^2 + 4x + 6$

Problem 6(b)

Step1: Group like terms

$(x^2 + 5x^2) + (4x - 3x) + (-9 + 11)$

Step2: Combine like terms

$6x^2 + x + 2$

Problem 6(c)

Step1: Group like terms

$(x^3 - 7x^3) + 3x^2 + (-2x - 6x) + 1$

Step2: Combine like terms

$-6x^3 + 3x^2 - 8x + 1$

Problem 6(d)

Step1: Group like terms

$(8x^3 - 8x^3) + (-x^2 + 2x^2) + (14x + x) + (-5 + 2)$

Step2: Combine like terms

$x^2 + 15x - 3$

Problem 7(a)

Step1: Distribute the negative sign

$3x^2 - x + 7 - x^2 + 5x - 4$

Step2: Group like terms

$(3x^2 - x^2) + (-x + 5x) + (7 - 4)$

Step3: Combine like terms

$2x^2 + 4x + 3$

Problem 7(b)

Step1: Distribute the negative sign

$-x^2 + 6x - 2 - 2x^2 - 9x + 7$

Step2: Group like terms

$(-x^2 - 2x^2) + (6x - 9x) + (-2 + 7)$

Step3: Combine like terms

$-3x^2 - 3x + 5$

Problem 7(c)

Step1: Distribute the negative sign

$8x + 4x^2 - 5 + 2x^2 - 4$

Step2: Group like terms

$(4x^2 + 2x^2) + 8x + (-5 - 4)$

Step3: Combine like terms

$6x^2 + 8x - 9$

Problem 7(d)

Step1: Distribute the negative sign

$x^3 - 6x^2 + 2x + 11 - 9x - 13 + 4x^2 - 2x^3$

Step2: Group like terms

$(x^3 - 2x^3) + (-6x^2 + 4x^2) + (2x - 9x) + (11 - 13)$

Step3: Combine like terms

$-x^3 - 2x^2 - 7x - 2$

Problem 8 (Given $f(x)=x^3-2x^2+8x-1$, $g(x)=7x^2-4x+3$)

Part (a) $f(x)+g(x)$

Step1: Substitute polynomials

$(x^3-2x^2+8x-1)+(7x^2-4x+3)$

Step2: Group like terms

$x^3 + (-2x^2+7x^2)+(8x-4x)+(-1+3)$

Step3: Combine like terms

$x^3 + 5x^2 + 4x + 2$

Part (b) $f(x)-g(x)$

Step1: Substitute polynomials, distribute negative

$(x^3-2x^2+8x-1)-(7x^2-4x+3)=x^3-2x^2+8x-1-7x^2+4x-3$

Step2: Group like terms

$x^3 + (-2x^2-7x^2)+(8x+4x)+(-1-3)$

Step3: Combine like terms

$x^3 - 9x^2 + 12x - 4$

Problem 9

Step1: Parallel integer-polynomial logic

An integer + integer = integer → polynomial + polynomial = polynomial

Step2: Formulate the statement

A polynomial added to a polynomial gives a polynomial.

Answer:

1. (a) $2x^2 + 4x + 6$

(b) $6x^2 + x + 2$
(c) $-6x^3 + 3x^2 - 8x + 1$
(d) $x^2 + 15x - 3$

1. (a) $2x^2 + 4x + 3$

(b) $-3x^2 - 3x + 5$
(c) $6x^2 + 8x - 9$
(d) $-x^3 - 2x^2 - 7x - 2$

1. (a) $x^3 + 5x^2 + 4x + 2$

(b) $x^3 - 9x^2 + 12x - 4$

1. A polynomial added to a polynomial gives a polynomial.