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(b) What should be subtracted from the sum of [tex]f(x) = 4x^2 + 5x + 6[/tex] and [tex]g(x) = x^2 + 3x + 4[/tex]?

a) If [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) + q(x) - r(x)[/tex].

b) If [tex]p(x) = x^2 + 2x + 3[/tex], [tex]q(x) = x^2 + 5x + 6[/tex], and [tex]r(x) = 4x^2 + 10x + 6[/tex], then find [tex][p(x) + q(x)] - r(x)[/tex].



Answer :

GinnyAnswer
Certainly! Let's solve part (b) step-by-step.

Given:
[tex]\[ p(x) = x^2 + 2x + 3 \][/tex]
[tex]\[ q(x) = x^2 + 5x + 6 \][/tex]
[tex]\[ r(x) = 4x^2 + 10x + 6 \][/tex]

We want to find [tex]\( (p(x) + q(x)) - r(x) \)[/tex].

Step 1: Calculate [tex]\( p(x) + q(x) \)[/tex]

[tex]\[ p(x) + q(x) = (x^2 + 2x + 3) + (x^2 + 5x + 6) \][/tex]

Combine like terms:

[tex]\[ = x^2 + x^2 + 2x + 5x + 3 + 6 \][/tex]
[tex]\[ = 2x^2 + 7x + 9 \][/tex]

So, [tex]\( p(x) + q(x) = 2x^2 + 7x + 9 \)[/tex].

Step 2: Subtract [tex]\( r(x) \)[/tex] from [tex]\( p(x) + q(x) \)[/tex]

[tex]\[ (p(x) + q(x)) - r(x) = (2x^2 + 7x + 9) - (4x^2 + 10x + 6) \][/tex]

Distribute the minus sign and then combine like terms:

[tex]\[ = 2x^2 + 7x + 9 - 4x^2 - 10x - 6 \][/tex]
[tex]\[ = (2x^2 - 4x^2) + (7x - 10x) + (9 - 6) \][/tex]
[tex]\[ = -2x^2 - 3x + 3 \][/tex]

So the result is:
[tex]\[ (p(x) + q(x)) - r(x) = -2x^2 - 3x + 3 \][/tex]

Therefore, the polynomial that should be subtracted from the sum of [tex]\( p(x) \)[/tex] and [tex]\( q(x) \)[/tex] to get the result is [tex]\( -2x^2 - 3x + 3 \)[/tex].

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