Math Problem Statement

Considera los siguientes polinomios: P(x) = 3x^2 + 5x - 2 Q(x) = -2x^2 + 4 S(x) = 2x^2 R(x) = 5x - 6 Realiza las siguientes multiplicaciones y expresa el resultado en forma de polinomio ordenado: 1. P(x) * S(x) 2. Q(x) * S(x) 3. Q(x) * R(x) 4. P(x) * Q(x) 5. P(x) * R(x)

Solution

Let's work through the multiplication of the polynomials given in the image.

The polynomials are:

P(x)=3x2+5x−2P(x) = 3x^2 + 5x - 2 Q(x)=−2x2+4Q(x) = -2x^2 + 4 S(x)=2x2S(x) = 2x^2 R(x)=5x−6R(x) = 5x - 6

Now, let's perform the requested multiplications.

1. P(x)×S(x)P(x) \times S(x)

P(x)×S(x)=(3x2+5x−2)×2x2P(x) \times S(x) = (3x^2 + 5x - 2) \times 2x^2

We distribute 2x22x^2 across each term of P(x)P(x):

=2x2(3x2)+2x2(5x)+2x2(−2)= 2x^2(3x^2) + 2x^2(5x) + 2x^2(-2) =6x4+10x3−4x2= 6x^4 + 10x^3 - 4x^2

Thus, the result is:

P(x)×S(x)=6x4+10x3−4x2P(x) \times S(x) = 6x^4 + 10x^3 - 4x^2

2. Q(x)×S(x)Q(x) \times S(x)

Q(x)×S(x)=(−2x2+4)×2x2Q(x) \times S(x) = (-2x^2 + 4) \times 2x^2

We distribute 2x22x^2 across each term of Q(x)Q(x):

=2x2(−2x2)+2x2(4)= 2x^2(-2x^2) + 2x^2(4) =−4x4+8x2= -4x^4 + 8x^2

Thus, the result is:

Q(x)×S(x)=−4x4+8x2Q(x) \times S(x) = -4x^4 + 8x^2

3. Q(x)×R(x)Q(x) \times R(x)

Q(x)×R(x)=(−2x2+4)×(5x−6)Q(x) \times R(x) = (-2x^2 + 4) \times (5x - 6)

We use the distributive property:

=(−2x2)(5x)+(−2x2)(−6)+(4)(5x)+(4)(−6)= (-2x^2)(5x) + (-2x^2)(-6) + (4)(5x) + (4)(-6) =−10x3+12x2+20x−24= -10x^3 + 12x^2 + 20x - 24

Thus, the result is:

Q(x)×R(x)=−10x3+12x2+20x−24Q(x) \times R(x) = -10x^3 + 12x^2 + 20x - 24

4. P(x)×Q(x)P(x) \times Q(x)

P(x)×Q(x)=(3x2+5x−2)×(−2x2+4)P(x) \times Q(x) = (3x^2 + 5x - 2) \times (-2x^2 + 4)

We distribute each term of P(x)P(x) over Q(x)Q(x):

=(3x2)(−2x2)+(3x2)(4)+(5x)(−2x2)+(5x)(4)+(−2)(−2x2)+(−2)(4)= (3x^2)(-2x^2) + (3x^2)(4) + (5x)(-2x^2) + (5x)(4) + (-2)(-2x^2) + (-2)(4) =−6x4+12x2−10x3+20x+4x2−8= -6x^4 + 12x^2 - 10x^3 + 20x + 4x^2 - 8

Combining like terms:

=−6x4−10x3+16x2+20x−8= -6x^4 - 10x^3 + 16x^2 + 20x - 8

Thus, the result is:

P(x)×Q(x)=−6x4−10x3+16x2+20x−8P(x) \times Q(x) = -6x^4 - 10x^3 + 16x^2 + 20x - 8

5. P(x)×R(x)P(x) \times R(x)

P(x)×R(x)=(3x2+5x−2)×(5x−6)P(x) \times R(x) = (3x^2 + 5x - 2) \times (5x - 6)

We distribute each term of P(x)P(x) over R(x)R(x):

=(3x2)(5x)+(3x2)(−6)+(5x)(5x)+(5x)(−6)+(−2)(5x)+(−2)(−6)= (3x^2)(5x) + (3x^2)(-6) + (5x)(5x) + (5x)(-6) + (-2)(5x) + (-2)(-6) =15x3−18x2+25x2−30x−10x+12= 15x^3 - 18x^2 + 25x^2 - 30x - 10x + 12

Combining like terms:

=15x3+7x2−40x+12= 15x^3 + 7x^2 - 40x + 12

Thus, the result is:

P(x)×R(x)=15x3+7x2−40x+12P(x) \times R(x) = 15x^3 + 7x^2 - 40x + 12

Would you like further details or have any questions about these calculations?

Here are five related questions you might find interesting:

  1. How do you perform polynomial division for the given polynomials?
  2. Can you graph the resulting polynomials from the multiplications?
  3. How would the behavior of the functions change if we added higher-degree terms?
  4. What are the roots of P(x)P(x), Q(x)Q(x), and other polynomials involved?
  5. Can you factor the resulting polynomials?

Tip: Always simplify polynomials by combining like terms after performing operations like multiplication or addition.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Polynomials
Multiplication of Polynomials

Formulas

Distributive Property: (a + b)(c + d) = ac + ad + bc + bd
Combining Like Terms: ax^n + bx^n = (a + b)x^n

Theorems

Distributive Law of Multiplication over Addition

Suitable Grade Level

Grades 10-12

Related Recommendation

Multiplying Polynomials P(x) = 3x^3 − 2x^2 + x − 5 and Q(x) = x^2 + x − 2

Polynomial Multiplication of p(x) and q(x) Using Distributive Property

Polynomial Addition: Find P(x) + Q(x) for P(x) = x^4 + x^2 - 4x - 3 and Q(x) = 3x^3 - 3x + 2

Polynomial Multiplication: Solve P(x) = 5x and R(x) = 5x - 8

How to Add Polynomials: P(x) = x^4 + x^2 - 4x - 3 and Q(x) = 3x^3 - 3x + 2