Answered

Subtract the second polynomial from the first and find the degree of the resulting polynomial.

1. [tex]\( p(x)=3x^3+5x-4 \)[/tex] and [tex]\( q(x)=6x^4-x+2 \)[/tex]
2. [tex]\( p(y)=6y^2+y-2 \)[/tex] and [tex]\( q(y)=-8y^3+3y^2+7 \)[/tex]
3. [tex]\( p(x)=8x^3+6x^2+5x+2 \)[/tex] and [tex]\( q(x)=8x^3-7x^2+4x+2 \)[/tex]
4. [tex]\( p(y)=y^3+3y^2+3y+1 \)[/tex] and [tex]\( q(y)=y^3-3y^2+3y-1 \)[/tex]
5. [tex]\( p(x)=x^3+3x^2+x+1 \)[/tex] and [tex]\( q(x)=x^4+5x^3+6x^2+5x+2 \)[/tex]
6. [tex]\( p(y)=y^4-4y^3+6y^2-4y+1 \)[/tex] and [tex]\( q(y)=y^2+2y+1 \)[/tex]
7. [tex]\( p(x)=x^4+2x^2+1 \)[/tex] and [tex]\( q(x)=x^4-2x^2+1 \)[/tex]
8. [tex]\( p(x)=11x^2-5x+7 \)[/tex], [tex]\( q(x)=13x^2+5x-9 \)[/tex], and [tex]\( r(x)=3x^2-6x+1 \)[/tex], then find [tex]\( p(x) \)[/tex]



Answer :

GinnyAnswer
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].