Certainly! Let's tackle each part of the question step-by-step:
### Question 10
Given Polynomial: [tex]\( x^3 - 64 \)[/tex]
1. Degree of the Polynomial:
- The degree of a polynomial is the highest power of the variable in the polynomial.
- In the polynomial [tex]\( x^3 - 64 \)[/tex], the highest power of the variable [tex]\( x \)[/tex] is 3.
- Therefore, the degree of the polynomial is 3.
2. Number of Terms in the Polynomial:
- The number of terms in a polynomial is simply the count of the different terms separated by [tex]\( + \)[/tex] or [tex]\( - \)[/tex] signs.
- In the polynomial [tex]\( x^3 - 64 \)[/tex], there are two terms: [tex]\( x^3 \)[/tex] and [tex]\( -64 \)[/tex].
- Therefore, the polynomial has 2 terms.
So, the polynomial [tex]\( x^3 - 64 \)[/tex] is of degree 3 and has 2 terms.
### Question 11
Given Polynomial Equation: [tex]\( x^2 - 3 x^2 + x - 3 = 4 \)[/tex]
1. Simplify the Given Polynomial Equation:
- First, rewrite the equation in a standard polynomial form.
- [tex]\( x^2 - 3 x^2 + x - 3 - 4 = 0 \)[/tex]
- Combine like terms:
- [tex]\( x^2 - 3x^2 = -2x^2 \)[/tex]
- The equation simplifies to [tex]\( -2x^2 + x - 7 = 0 \)[/tex].
2. Checking if [tex]\( x = 3 \)[/tex] is a Zero of the Polynomial:
- Substitute [tex]\( x = 3 \)[/tex] into the simplified polynomial [tex]\( -2x^2 + x - 7 \)[/tex].
- Calculate the polynomial value at [tex]\( x = 3 \)[/tex]:
[tex]\[
-2(3)^2 + 3 - 7
\][/tex]
[tex]\[
-2(9) + 3 - 7
\][/tex]
[tex]\[
-18 + 3 - 7
\][/tex]
[tex]\[
-22
\][/tex]
- So, the value of the polynomial at [tex]\( x = 3 \)[/tex] is [tex]\( -22 \)[/tex].
3. Determine if it's a Zero:
- For [tex]\( x = 3 \)[/tex] to be a zero of the polynomial, the polynomial should equal 0 when [tex]\( x = 3 \)[/tex].
- Since the calculated value is [tex]\( -22 \)[/tex] (not 0), [tex]\( x = 3 \)[/tex] is not a zero of the polynomial [tex]\( -2x^2 + x - 7 \)[/tex].
Hence, [tex]\( x = 3 \)[/tex] is not a zero of the polynomial [tex]\( x^2 - 3 x^2 + x - 3 = 4 \)[/tex].
