Solve polynomial addition and subtraction problems as provided in the exercise.
Thank you for providing the image. Let me analyze it and extract the mathematical content to solve the problems.The image contains two sets of polynomial operations. Here are the tasks for both:
Exercise 22 : Add the polynomials P ( x ) P(x) P ( x ) Q ( x ) Q(x) Q ( x ) Exercise 23 : Subtract Q ( x ) Q(x) Q ( x ) P ( x ) P(x) P ( x ) I'll solve these step by step, starting with Exercise 22 .
Exercise 22: Suma els polinomis següents
(a) P ( x ) = − x 2 + 3 x P(x) = -x^2 + 3x P ( x ) = − x 2 + 3 x Q ( x ) = 5 x − 2 Q(x) = 5x - 2 Q ( x ) = 5 x − 2 P ( x ) + Q ( x ) = ( − x 2 ) + ( 3 x + 5 x ) + ( − 2 ) = − x 2 + 8 x − 2 P(x) + Q(x) = (-x^2) + (3x + 5x) + (-2) = -x^2 + 8x - 2 P ( x ) + Q ( x ) = ( − x 2 ) + ( 3 x + 5 x ) + ( − 2 ) = − x 2 + 8 x − 2
(b) P ( x ) = − 2 x 2 + 4 x P(x) = -2x^2 + 4x P ( x ) = − 2 x 2 + 4 x Q ( x ) = 5 x 2 + 2 x Q(x) = 5x^2 + 2x Q ( x ) = 5 x 2 + 2 x P ( x ) + Q ( x ) = ( − 2 x 2 + 5 x 2 ) + ( 4 x + 2 x ) = 3 x 2 + 6 x P(x) + Q(x) = (-2x^2 + 5x^2) + (4x + 2x) = 3x^2 + 6x P ( x ) + Q ( x ) = ( − 2 x 2 + 5 x 2 ) + ( 4 x + 2 x ) = 3 x 2 + 6 x
(c) P ( x ) = − x 2 + x 3 − x P(x) = -x^2 + x^3 - x P ( x ) = − x 2 + x 3 − x Q ( x ) = 2 x − 2 x 2 Q(x) = 2x - 2x^2 Q ( x ) = 2 x − 2 x 2 P ( x ) + Q ( x ) = x 3 + ( − x 2 − 2 x 2 ) + ( − x + 2 x ) = x 3 − 3 x 2 + x P(x) + Q(x) = x^3 + (-x^2 - 2x^2) + (-x + 2x) = x^3 - 3x^2 + x P ( x ) + Q ( x ) = x 3 + ( − x 2 − 2 x 2 ) + ( − x + 2 x ) = x 3 − 3 x 2 + x
(d) P ( x ) = 1 + x + x 2 P(x) = 1 + x + x^2 P ( x ) = 1 + x + x 2 Q ( x ) = x − 1 Q(x) = x - 1 Q ( x ) = x − 1 P ( x ) + Q ( x ) = ( 1 − 1 ) + x + x + x 2 = x 2 + 2 x P(x) + Q(x) = (1 - 1) + x + x + x^2 = x^2 + 2x P ( x ) + Q ( x ) = ( 1 − 1 ) + x + x + x 2 = x 2 + 2 x
(e) P ( x ) = 2 x + x 2 − 5 P(x) = 2x + x^2 - 5 P ( x ) = 2 x + x 2 − 5 Q ( x ) = x + 3 x 3 Q(x) = x + 3x^3 Q ( x ) = x + 3 x 3 P ( x ) + Q ( x ) = 3 x 3 + x 2 + ( 2 x + x ) − 5 = 3 x 3 + x 2 + 3 x − 5 P(x) + Q(x) = 3x^3 + x^2 + (2x + x) - 5 = 3x^3 + x^2 + 3x - 5 P ( x ) + Q ( x ) = 3 x 3 + x 2 + ( 2 x + x ) − 5 = 3 x 3 + x 2 + 3 x − 5
(f) P ( x ) = 2 x 2 − 4 x P(x) = 2x^2 - 4x P ( x ) = 2 x 2 − 4 x Q ( x ) = 3 x 4 − 2 x 2 Q(x) = 3x^4 - 2x^2 Q ( x ) = 3 x 4 − 2 x 2 P ( x ) + Q ( x ) = 3 x 4 + ( 2 x 2 − 2 x 2 ) − 4 x = 3 x 4 − 4 x P(x) + Q(x) = 3x^4 + (2x^2 - 2x^2) - 4x = 3x^4 - 4x P ( x ) + Q ( x ) = 3 x 4 + ( 2 x 2 − 2 x 2 ) − 4 x = 3 x 4 − 4 x
(g) P ( x ) = − 2 x 2 + 6 x P(x) = -2x^2 + 6x P ( x ) = − 2 x 2 + 6 x Q ( x ) = 2 x 2 − 6 x Q(x) = 2x^2 - 6x Q ( x ) = 2 x 2 − 6 x P ( x ) + Q ( x ) = ( − 2 x 2 + 2 x 2 ) + ( 6 x − 6 x ) = 0 P(x) + Q(x) = (-2x^2 + 2x^2) + (6x - 6x) = 0 P ( x ) + Q ( x ) = ( − 2 x 2 + 2 x 2 ) + ( 6 x − 6 x ) = 0
Exercise 23: Calcula P ( x ) − Q ( x ) P(x) - Q(x) P ( x ) − Q ( x )
(a) P ( x ) = 5 x 4 + 3 x 3 P(x) = 5x^4 + 3x^3 P ( x ) = 5 x 4 + 3 x 3 Q ( x ) = 5 x − 2 x 3 Q(x) = 5x - 2x^3 Q ( x ) = 5 x − 2 x 3 P ( x ) − Q ( x ) = 5 x 4 + ( 3 x 3 + 2 x 3 ) − 5 x = 5 x 4 + 5 x 3 − 5 x P(x) - Q(x) = 5x^4 + (3x^3 + 2x^3) - 5x = 5x^4 + 5x^3 - 5x P ( x ) − Q ( x ) = 5 x 4 + ( 3 x 3 + 2 x 3 ) − 5 x = 5 x 4 + 5 x 3 − 5 x
(b) P ( x ) = x 3 + 3 x − 7 P(x) = x^3 + 3x - 7 P ( x ) = x 3 + 3 x − 7 Q ( x ) = 5 x − 2 x 2 Q(x) = 5x - 2x^2 Q ( x ) = 5 x − 2 x 2 P ( x ) − Q ( x ) = x 3 − 2 x 2 + ( 3 x − 5 x ) − 7 = x 3 − 2 x 2 − 2 x − 7 P(x) - Q(x) = x^3 - 2x^2 + (3x - 5x) - 7 = x^3 - 2x^2 - 2x - 7 P ( x ) − Q ( x ) = x 3 − 2 x 2 + ( 3 x − 5 x ) − 7 = x 3 − 2 x 2 − 2 x − 7
(c) P ( x ) = 13 x 2 − x P(x) = 13x^2 - x P ( x ) = 13 x 2 − x Q ( x ) = 13 x + x 2 Q(x) = 13x + x^2 Q ( x ) = 13 x + x 2 P ( x ) − Q ( x ) = ( 13 x 2 − x 2 ) − ( x + 13 x ) = 12 x 2 − 14 x P(x) - Q(x) = (13x^2 - x^2) - (x + 13x) = 12x^2 - 14x P ( x ) − Q ( x ) = ( 13 x 2 − x 2 ) − ( x + 13 x ) = 12 x 2 − 14 x
(d) P ( x ) = 9 x 2 + 8 P(x) = 9x^2 + 8 P ( x ) = 9 x 2 + 8 Q ( x ) = 7 x − 6 Q(x) = 7x - 6 Q ( x ) = 7 x − 6 P ( x ) − Q ( x ) = 9 x 2 − 7 x + ( 8 + 6 ) = 9 x 2 − 7 x + 14 P(x) - Q(x) = 9x^2 - 7x + (8 + 6) = 9x^2 - 7x + 14 P ( x ) − Q ( x ) = 9 x 2 − 7 x + ( 8 + 6 ) = 9 x 2 − 7 x + 14
(e) P ( x ) = 5 x 2 − 4 x P(x) = 5x^2 - 4x P ( x ) = 5 x 2 − 4 x Q ( x ) = 5 x + x 2 + 8 Q(x) = 5x + x^2 + 8 Q ( x ) = 5 x + x 2 + 8 P ( x ) − Q ( x ) = ( 5 x 2 − x 2 ) − 4 x − 5 x − 8 = 4 x 2 − 9 x − 8 P(x) - Q(x) = (5x^2 - x^2) - 4x - 5x - 8 = 4x^2 - 9x - 8 P ( x ) − Q ( x ) = ( 5 x 2 − x 2 ) − 4 x − 5 x − 8 = 4 x 2 − 9 x − 8
(f) P ( x ) = − 2 x 2 + 3 x 3 P(x) = -2x^2 + 3x^3 P ( x ) = − 2 x 2 + 3 x 3 Q ( x ) = 2 x 3 − 3 x 2 Q(x) = 2x^3 - 3x^2 Q ( x ) = 2 x 3 − 3 x 2 P ( x ) − Q ( x ) = ( 3 x 3 − 2 x 3 ) + ( − 2 x 2 + 3 x 2 ) = x 3 + x 2 P(x) - Q(x) = (3x^3 - 2x^3) + (-2x^2 + 3x^2) = x^3 + x^2 P ( x ) − Q ( x ) = ( 3 x 3 − 2 x 3 ) + ( − 2 x 2 + 3 x 2 ) = x 3 + x 2
(g) P ( x ) = 4 x 3 − 3 x 2 + x P(x) = 4x^3 - 3x^2 + x P ( x ) = 4 x 3 − 3 x 2 + x Q ( x ) = − x + x 2 + x 4 Q(x) = -x + x^2 + x^4 Q ( x ) = − x + x 2 + x 4 P ( x ) − Q ( x ) = − x 4 + 4 x 3 + ( − 3 x 2 − x 2 ) + ( x + x ) = − x 4 + 4 x 3 − 4 x 2 + 2 x P(x) - Q(x) = -x^4 + 4x^3 + (-3x^2 - x^2) + (x + x) = -x^4 + 4x^3 - 4x^2 + 2x P ( x ) − Q ( x ) = − x 4 + 4 x 3 + ( − 3 x 2 − x 2 ) + ( x + x ) = − x 4 + 4 x 3 − 4 x 2 + 2 x
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Related Questions:
What is the general rule for adding polynomials?
How do you organize terms when subtracting polynomials?
Why is it important to match like terms in polynomial operations?
What happens if terms cancel out entirely, as in Exercise 22(g)?
How can the degree of a polynomial affect addition or subtraction?
Tip : Always align similar terms (e.g., x 2 , x 3 x^2, x^3 x 2 , x 3