Answer:
a. see below
b. -10x^6 +24x^4 -43x^3 -8x^2 +30x -28
c. no. One has a positive x^6 term; the other a negative x^6 term.
Step-by-step explanation:
a) We want to multiply ...
I find it convenient to do this by looking at the way product terms can be formed.* Product terms will have a variable portion that ranges from x^6 down to 1 (=x^0).
The x^6 term will be the product of x^3 terms. It will have a coefficient of (-5)(2) = -10.
There is no way to make an x^5 term from these factors.
The x^4 term will be the sum of products of x^3 and x terms. It will have a coefficient of (-5)(-4) +(2)(2) = 24.
The x^3 term will be the sum of products of x^3 terms and constants. It will have a coefficient of (-5)(7) +(2)(-4) = -35 -8 = -43.
The x^2 term will be the product of x terms. It will have a coefficient of (2)(-4) = -8.
The x term will be the sum of products of x and constant terms. It will have a coefficient of (2)(7) +(-4)(-4) = 14 +16 = 30.
The constant term will be the product of the constants: (-4)(7) = -28.
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b) The product we're looking for is ...
-10x^6 +24x^4 -43x^3 -8x^2 +30x -28
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c) The product of (2x^3 -4x +7) and (5x^3 +2x -4) will have a highest-power term of (2x^3)(5x^3) = 10x^6. This is different from the highest-power term of the product we just calculated. The second product cannot be the same as the first.
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* Note that had there been x^2 terms in each expression, the ways of forming x^3 product terms would include the products of x and x^2 terms as well as the products of x^3 and constant terms. Likewise, the x^4 product term would include the product of x^2 terms. When using this method, one needs to look at all the possibilities.
When coefficients are written in standard order and 0s are filled in for missing terms, the pattern of product sums is a pattern of Xs that is reasonably easy to see and remember.
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If you cannot see or don't understand the above method of multiplying polynomials, you can always make use of the distributive property and collect terms in the usual way.
(-5x^3 +2x -4)( ) = -5x^3( ) +2x( ) -4( )
