Question

2 fill in the blank 7 points
multiply the polynomials. $(x + 5)(x^2 - 7x + 4)$
answer: $ax^b + cx^d + ex^f + g$
$a = $ type your answer...
$b = $ type your answer...
$c = $ type your answer...
$d = $ type your answer...
$e = $ type your answer...
$f = $ type your answer...
$g = $ type your answer...

Explanation:

Step1: Apply distributive property (FOIL for polynomials)

Multiply \( x \) by each term in \( x^2 - 7x + 4 \): \( x \cdot x^2 = x^3 \), \( x \cdot (-7x) = -7x^2 \), \( x \cdot 4 = 4x \).
Then multiply \( 5 \) by each term in \( x^2 - 7x + 4 \): \( 5 \cdot x^2 = 5x^2 \), \( 5 \cdot (-7x) = -35x \), \( 5 \cdot 4 = 20 \).

Step2: Combine like terms

For \( x^3 \): only \( x^3 \).
For \( x^2 \): \( -7x^2 + 5x^2 = -2x^2 \).
For \( x \): \( 4x - 35x = -31x \).
For the constant term: \( 20 \).
So the expanded form is \( x^3 - 2x^2 - 31x + 20 \).

Step3: Identify coefficients and exponents

Compare with \( ax^b + cx^d + ex^f + g \):

• \( a = 1 \), \( b = 3 \) (from \( x^3 \))
• \( c = -2 \), \( d = 2 \) (from \( -2x^2 \))
• \( e = -31 \), \( f = 1 \) (from \( -31x = -31x^1 \))
• \( g = 20 \) (constant term)

Answer:

\( a = 1 \), \( b = 3 \), \( c = -2 \), \( d = 2 \), \( e = -31 \), \( f = 1 \), \( g = 20 \)