Answer:
d. x^3 − 2x^2 − 21x + 12
Step-by-step explanation:
You only need to look at the x^2 term to determine the correct choice. That term will be the sum of two products: 4(x^2) + x(-6x) = -2x^2.
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In general, you "simplify" such an expression by using the distributive property as many times as necessary.
(x + 4)(x^2 -6x +3)
= x(x^2 -6x +3) + 4(x^2 -6x +3)
= (x^3 -6x^2 +3x) +(4x^2 -24x +12)
= x^3 +(-6+4)x^2 +(3-24)x +12
= x^3 -2x^2 -21x +12
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You can write the coefficients of x^2, x, and constant in two rows:
0 1 4
1 -6 3
Now, the coefficients of the product can be found by considering the sums of various "X" patterns. You can write them down using mental arithmetic, working left to right.
The x^4 term is the product of the leftmost coefficients: 0·1 = 0
The x^3 term is the product of pairs of the left 2 coefficients: 0·(-6) +1·1 = 1
The x^2 term is the product of extreme left/right coefficients together with the center two: 0·3 +4·1 +1(-6) = -2
The x term is the product of pairs of the right 2 coefficients: 1·3 +4(-6) = -21
The constant term is the product of the extreme right coefficients: 4·3 = 12
So, the final product is ...
0x^4 +1x^3 -2x^2 -21x +12 = x^3 -2x^2 -21x +12
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This method of working out products of polynomials or numbers is taught elsewhere in the world. It ensures that each term in one factor is multiplied by every term in the other factor, essentially as the distributive property does. By grouping the sums of products this way, we get to the answer more quickly and can write it with all the "work" being mental.
