To solve the problem, we need to subtract the given polynomials and find the degree of the resulting polynomial. We have:
[tex]\[ p(x) = 11x^2 - 5x + 7 \][/tex]
[tex]\[ q(x) = 13x^2 + 5x - 9 \][/tex]
[tex]\[ r(x) = 3x^2 - 6x + 1 \][/tex]
We perform the following steps:
1. Subtract [tex]\( q(x) \)[/tex] from [tex]\( p(x) \)[/tex]:
[tex]\[ (11x^2 - 5x + 7) - (13x^2 + 5x - 9) \][/tex]
Let's distribute the negative sign through [tex]\( q(x) \)[/tex]:
[tex]\[ 11x^2 - 5x + 7 - 13x^2 - 5x + 9 \][/tex]
Combining like terms:
[tex]\[ (11x^2 - 13x^2) + (-5x - 5x) + (7 + 9) \][/tex]
[tex]\[ -2x^2 - 10x + 16 \][/tex]
Thus, the resulting polynomial after subtracting [tex]\( q(x) \)[/tex] from [tex]\( p(x) \)[/tex] is:
[tex]\[ p(x) - q(x) = -2x^2 - 10x + 16 \][/tex]
2. Subtract [tex]\( r(x) \)[/tex] from the result:
[tex]\[ (-2x^2 - 10x + 16) - (3x^2 - 6x + 1) \][/tex]
Let's distribute the negative sign through [tex]\( r(x) \)[/tex]:
[tex]\[ -2x^2 - 10x + 16 - 3x^2 + 6x - 1 \][/tex]
Combining like terms:
[tex]\[ (-2x^2 - 3x^2) + (-10x + 6x) + (16 - 1) \][/tex]
[tex]\[ -5x^2 - 4x + 15 \][/tex]
Thus, the final polynomial after subtracting [tex]\( r(x) \)[/tex] from [tex]\( (p(x) - q(x)) \)[/tex] is:
[tex]\[ (p(x) - q(x)) - r(x) = -5x^2 - 4x + 15 \][/tex]
3. Find the degree of the resulting polynomial:
The degree of a polynomial is the highest power of [tex]\( x \)[/tex] with a non-zero coefficient in the polynomial. For the polynomial [tex]\( -5x^2 - 4x + 15 \)[/tex], the highest power of [tex]\( x \)[/tex] is 2. Therefore, the degree of this polynomial is:
[tex]\[ \text{Degree} = 2 \][/tex]
Hence, the final degree of the polynomial [tex]\( (p(x) - q(x)) - r(x) \)[/tex] is [tex]\( 2 \)[/tex].
