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Answer :
To find the zeros of the polynomial function f(x) = x⁴ - 4x³ + 23x² + 62x + 34, we find the values of x for which the function equals zero (f(x) = 0). x = -1 is a zero of polynomial function.
One way to approach this is by using the Rational Root Theorem to test the potential rational zeros. The Rational Root Theorem states that if a rational number p/q is a root of a polynomial with integer coefficients, then p is a factor of constant term and q is a factor of the leading coefficient (1 in this case).
The factors of 34 are ±1, ±2, ±17, and ±34.
The factors of 1 are ±1.
So, potential rational zeros are:
±1, ±2, ±17, ±34
We now use these potential zeros to check if they are indeed zeros of polynomial function. By evaluating the function at each potential zero, we determine if it equals zero.
f(1) = (1)⁴ - 4(1)³ + 23(1)² + 62(1) + 34
= 116
≠ 0
f(-1) = (-1)⁴ - 4(-1)³ + 23(-1)² + 62(-1) + 34
= 0 ✓
f(2) = (2)⁴ - 4(2)³ + 23(2)² + 62(2) + 34
= 234
≠ 0
f(-2) = (-2)⁴ - 4(-2)³ + 23(-2)² + 62(-2) + 34
= 50
≠ 0
f(17) = (17)⁴ - 4(17)³ + 23(17)² + 62(17) + 34
= 242980
≠ 0
f(-17) = (-17)⁴ - 4(-17)³ + 23(-17)² + 62(-17) + 34
= 301608
≠ 0
f(34) = (34)⁴ - 4(34)³ + 23(34)² + 62(34) + 34
= 1194010
≠ 0
f(-34) = (-34)⁴ - 4(-34)³ + 23(-34)² + 62(-34) + 34
= 1362638
≠ 0
From the tests, we see that f(-1) = 0. Therefore, x = -1 is a zero of polynomial function.
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Note: The complete question is :
Question: The polynomial function is f(x)=x⁴-4x³+23x²+62x+34. Find the zeros of the polynomial function.
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