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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