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Answer :
To find all rational zeros of the polynomial function [tex]\( P(x) = x^7 + 7x^6 + 21x^5 + 35x^4 + 35x^3 + 21x^2 + 7x + 1 \)[/tex], we can use the Rational Root Theorem and other algebraic techniques. Here's how you can approach this:
1. Identify Possible Rational Zeros: The Rational Root Theorem states that any rational zero, in its simplest form [tex]\( \frac{p}{q} \)[/tex], is such that [tex]\( p \)[/tex] is a factor of the constant term, and [tex]\( q \)[/tex] is a factor of the leading coefficient. For our polynomial, both the constant term and the leading coefficient are 1, so the rational zeros are factors of 1. Therefore, the possible rational zeros are [tex]\( \pm 1 \)[/tex].
2. Test the Possible Rational Zeros: Check if any of these candidates are actual zeros of the polynomial by substituting them into the polynomial function:
- Test [tex]\( x = 1 \)[/tex]:
[tex]\[
P(1) = 1^7 + 7 \times 1^6 + 21 \times 1^5 + 35 \times 1^4 + 35 \times 1^3 + 21 \times 1^2 + 7 \times 1 + 1 = 128
\][/tex]
Since [tex]\( P(1) \neq 0 \)[/tex], [tex]\( x = 1 \)[/tex] is not a zero.
- Test [tex]\( x = -1 \)[/tex]:
[tex]\[
P(-1) = (-1)^7 + 7 \times (-1)^6 + 21 \times (-1)^5 + 35 \times (-1)^4 + 35 \times (-1)^3 + 21 \times (-1)^2 + 7 \times (-1) + 1
\][/tex]
Simplifying this gives:
[tex]\[
-1 + 7 - 21 + 35 - 35 + 21 - 7 + 1 = 0
\][/tex]
Since [tex]\( P(-1) = 0 \)[/tex], [tex]\( x = -1 \)[/tex] is indeed a zero of the polynomial.
3. Conclusion: The only rational zero of the polynomial function [tex]\( P(x) \)[/tex] is [tex]\(-1\)[/tex].
Therefore, the rational zero of [tex]\( P(x) \)[/tex] is [tex]\(-1\)[/tex].
1. Identify Possible Rational Zeros: The Rational Root Theorem states that any rational zero, in its simplest form [tex]\( \frac{p}{q} \)[/tex], is such that [tex]\( p \)[/tex] is a factor of the constant term, and [tex]\( q \)[/tex] is a factor of the leading coefficient. For our polynomial, both the constant term and the leading coefficient are 1, so the rational zeros are factors of 1. Therefore, the possible rational zeros are [tex]\( \pm 1 \)[/tex].
2. Test the Possible Rational Zeros: Check if any of these candidates are actual zeros of the polynomial by substituting them into the polynomial function:
- Test [tex]\( x = 1 \)[/tex]:
[tex]\[
P(1) = 1^7 + 7 \times 1^6 + 21 \times 1^5 + 35 \times 1^4 + 35 \times 1^3 + 21 \times 1^2 + 7 \times 1 + 1 = 128
\][/tex]
Since [tex]\( P(1) \neq 0 \)[/tex], [tex]\( x = 1 \)[/tex] is not a zero.
- Test [tex]\( x = -1 \)[/tex]:
[tex]\[
P(-1) = (-1)^7 + 7 \times (-1)^6 + 21 \times (-1)^5 + 35 \times (-1)^4 + 35 \times (-1)^3 + 21 \times (-1)^2 + 7 \times (-1) + 1
\][/tex]
Simplifying this gives:
[tex]\[
-1 + 7 - 21 + 35 - 35 + 21 - 7 + 1 = 0
\][/tex]
Since [tex]\( P(-1) = 0 \)[/tex], [tex]\( x = -1 \)[/tex] is indeed a zero of the polynomial.
3. Conclusion: The only rational zero of the polynomial function [tex]\( P(x) \)[/tex] is [tex]\(-1\)[/tex].
Therefore, the rational zero of [tex]\( P(x) \)[/tex] is [tex]\(-1\)[/tex].
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