Sure! Let's tackle each part of the problem one by one.
### Part 8: Finding the quadratic polynomial
We are asked to find the quadratic polynomial whose sum of the zeroes is [tex]\(-5\)[/tex] and the product of the zeroes is [tex]\(6\)[/tex].
In general, for a quadratic polynomial [tex]\(ax^2 + bx + c\)[/tex]:
- The sum of the zeroes, [tex]\(\alpha + \beta\)[/tex], is given by [tex]\(-\frac{b}{a}\)[/tex]
- The product of the zeroes, [tex]\(\alpha \beta\)[/tex], is given by [tex]\(\frac{c}{a}\)[/tex]
Given:
- The sum of zeroes ([tex]\(\alpha + \beta\)[/tex]) = [tex]\(-5\)[/tex]
- The product of zeroes ([tex]\(\alpha \beta\)[/tex]) = [tex]\(6\)[/tex]
For a standard quadratic polynomial in the form [tex]\(x^2 + bx + c\)[/tex], we can directly match the coefficients:
- The sum of the zeroes introduces the term [tex]\(\sum{x}\)[/tex] to be [tex]\(-b\)[/tex], thus [tex]\(b = - (\text{sum of zeroes}) = -(-5) = 5\)[/tex].
- The product of the zeroes gives the constant term [tex]\(c\)[/tex], thus [tex]\(c = \text{product of zeroes} = 6\)[/tex].
Thus, the quadratic polynomial is:
[tex]\[ x^2 + 5x + 6 \][/tex]
### Part 9: Finding the value of [tex]\(m\)[/tex]
Now, we are given the polynomial [tex]\( p(x) = 4x^2 - 6x - m \)[/tex] and we need to find the value of [tex]\(m\)[/tex] such that this polynomial is exactly divisible by [tex]\(x - 3\)[/tex].
For a polynomial to be exactly divisible by another polynomial [tex]\(x - a\)[/tex], the remainder when the polynomial is divided by [tex]\(x - a\)[/tex] must be zero. This means that [tex]\(p(a) = 0\)[/tex].
Given that [tex]\(x - 3\)[/tex] is a factor of [tex]\(p(x)\)[/tex], we have:
[tex]\[ p(3) = 0 \][/tex]
Let's substitute [tex]\(x = 3\)[/tex] into the polynomial [tex]\( p(x) \)[/tex]:
[tex]\[ p(3) = 4(3)^2 - 6(3) - m \][/tex]
Calculating the terms inside:
[tex]\[ 4(3)^2 = 4 \cdot 9 = 36 \][/tex]
[tex]\[ -6(3) = -18 \][/tex]
So,
[tex]\[ p(3) = 36 - 18 - m = 0 \][/tex]
Simplifying this equation to solve for [tex]\(m\)[/tex]:
[tex]\[ 36 - 18 - m = 0 \][/tex]
[tex]\[ 18 - m = 0 \][/tex]
[tex]\[ m = 18 \][/tex]
Thus, the value of [tex]\(m\)[/tex] is:
[tex]\[ m = 18 \][/tex]
### Summary:
1. The quadratic polynomial whose sum of zeroes is [tex]\(-5\)[/tex] and product is [tex]\(6\)[/tex] is [tex]\(x^2 + 5x + 6\)[/tex].
2. The value of [tex]\(m\)[/tex] that makes the polynomial [tex]\(p(x) = 4x^2 - 6x - m\)[/tex] exactly divisible by [tex]\(x - 3\)[/tex] is [tex]\(18\)[/tex].
