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Answer :
Let's break down the polynomial [tex]\( k(x) = 45x^5 - 114x^4 + 26x^3 + 44x^2 + x - 2 \)[/tex] and answer each part of the question step-by-step:
1. Degree of [tex]\( k \)[/tex]:
- The degree of a polynomial is the highest power of [tex]\( x \)[/tex] with a non-zero coefficient. In [tex]\( k(x) \)[/tex], the highest power of [tex]\( x \)[/tex] is 5, so the degree is [tex]\( 5 \)[/tex].
2. Leading Coefficient of [tex]\( k \)[/tex]:
- The leading coefficient is the coefficient of the term with the highest power of [tex]\( x \)[/tex]. Here, the term is [tex]\( 45x^5 \)[/tex], so the leading coefficient is [tex]\( 45 \)[/tex].
3. End Behavior:
- For a polynomial, the end behavior depends on the leading term, which is [tex]\( 45x^5 \)[/tex] in this case.
- Since the degree (5) is odd and the leading coefficient (45) is positive, the end behavior is:
- Right hand end behavior: As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( k(x) \rightarrow \infty \)[/tex].
- Left hand end behavior: As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( k(x) \rightarrow -\infty \)[/tex].
4. Zeros of [tex]\( k \)[/tex]:
- The zeros are the values of [tex]\( x \)[/tex] that make [tex]\( k(x) = 0 \)[/tex]. From the solution, the zeros are [tex]\( x = -\frac{1}{3}, \frac{1}{5}, 1, \)[/tex] and [tex]\( 2 \)[/tex].
5. Factored Form:
- Writing the polynomial in factored form involves expressing it as a product of linear factors:
[tex]\[
k(x) = (x - 2)(x - 1)(3x + 1)^2(5x - 1)
\][/tex]
- This factored form reflects the zeros found, showing the multiplication of factors that yield zero at those points.
Now you have each part of the solution detailed and explained for the polynomial [tex]\( k(x) \)[/tex].
1. Degree of [tex]\( k \)[/tex]:
- The degree of a polynomial is the highest power of [tex]\( x \)[/tex] with a non-zero coefficient. In [tex]\( k(x) \)[/tex], the highest power of [tex]\( x \)[/tex] is 5, so the degree is [tex]\( 5 \)[/tex].
2. Leading Coefficient of [tex]\( k \)[/tex]:
- The leading coefficient is the coefficient of the term with the highest power of [tex]\( x \)[/tex]. Here, the term is [tex]\( 45x^5 \)[/tex], so the leading coefficient is [tex]\( 45 \)[/tex].
3. End Behavior:
- For a polynomial, the end behavior depends on the leading term, which is [tex]\( 45x^5 \)[/tex] in this case.
- Since the degree (5) is odd and the leading coefficient (45) is positive, the end behavior is:
- Right hand end behavior: As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( k(x) \rightarrow \infty \)[/tex].
- Left hand end behavior: As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( k(x) \rightarrow -\infty \)[/tex].
4. Zeros of [tex]\( k \)[/tex]:
- The zeros are the values of [tex]\( x \)[/tex] that make [tex]\( k(x) = 0 \)[/tex]. From the solution, the zeros are [tex]\( x = -\frac{1}{3}, \frac{1}{5}, 1, \)[/tex] and [tex]\( 2 \)[/tex].
5. Factored Form:
- Writing the polynomial in factored form involves expressing it as a product of linear factors:
[tex]\[
k(x) = (x - 2)(x - 1)(3x + 1)^2(5x - 1)
\][/tex]
- This factored form reflects the zeros found, showing the multiplication of factors that yield zero at those points.
Now you have each part of the solution detailed and explained for the polynomial [tex]\( k(x) \)[/tex].
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