High School

For the polynomial function [tex]f(x)=x^4+11x^3+21x^2-59x-70[/tex], we know [tex](x+1)[/tex] and [tex](x+7)[/tex] are factors. Find all of the linear factors.

Answer :

Final answer:

To find all of the linear factors for the polynomial function f(x)=x⁴+11x³+21x²-59x-70 given that (x+1) and (x+7) are already known factors, we can use polynomial long division to divide f(x) by (x+1) and (x+7) successively.

Explanation:

To find all of the linear factors for the polynomial function f(x)=x⁴+11x³+21x²-59x-70 given that (x+1) and (x+7) are already known factors, we can use polynomial long division to divide f(x) by (x+1) and (x+7) successively.

Dividing f(x) by (x+1) gives us a quotient of x³+10x²+11x-70 and a remainder of 0. This means that (x+1) is indeed a factor.

Next, we divide the quotient x³+10x²+11x-70 by (x+7). We find that the quotient is x²+3x-10 and the remainder is 0. Thus, (x+7) is also a factor.

Therefore, the linear factors of f(x) are (x+1) and (x+7).

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