Question

Solve the following polynomiasls.

P1 = 3x² - 2x + 1
P2 = 2x - 3

I) P1 + P2
II) P1 • P2
III) P1 - P2​

Answer 1

`\textsf{(i)}\quad P_1+P_2=3x^2-2`

`\textsf{(ii)}\quad P_1\cdot P_2=6x^2-13x^2+8x-3`

`\textsf{(iii)}\quad P_1-P_2=3x^2-4x+4`

Step-by-step explanation:

Given polynomials:

`P_1=3x^2-2x+1`

`P_2=2x-3`

Part (i)

To add the polynomials P₁ and P₂, simply combine like terms:

`\begin{aligned}P_1+P_2&=(3x^2-2x+1)+(2x-3)\\&=3x^2-2x+1+2x-3\\&=3x^2-2x+2x+1-3\\&=3x^2+1-3\\&=3x^2-2\end{aligned}`

Part (ii)

To multiply the polynomials P₁ and P₂. use the distributive property. This involves multiplying each term in P₁ by each term in P₂​ and then combining like terms:

`\begin{aligned}P_1\cdot P_2&=(3x^2-2x+1)(2x-3)\\&=3x^2(2x)+3x^2(-3)-2x(2x)-2x(-3)+1(2x)+1(-3)\\&=6x^3-9x^2-4x^2+6x+2x-3\\&=6x^2-13x^2+8x-3\end{aligned}`

Part (iii)

To subtract P₂ from P₁, distribute the negative sign to each term inside the second set of parentheses and then combine like terms:

`\begin{aligned}P_1-P_2&=(3x^2-2x+1)-(2x-3)\\&=3x^2-2x+1-2x+3\\&=3x^2-2x-2x+1+3\\&=3x^2-4x+4\end{aligned}`

Answer 2

The sum, product, and difference of the polynomials P1 and P2 were calculated by combining like terms, multiplying polynomials, and then simplifying the resulting expressions.

Step-by-step explanation:

To solve the given problems, we'll combine polynomials P1 and P2 in different ways as instructed. The polynomials are P1 = 3x² - 2x + 1 and P2 = 2x - 3.

I) P1 + P2

First, let's add P1 to P2: (3x² - 2x + 1) + (2x - 3) = 3x² + (-2x + 2x) + (1 - 3) = 3x² - 2. We've combined like terms where possible.

II) P1 • P2

Next, we'll find the product of P1 and P2: (3x² - 2x + 1)(2x - 3) = 3x²(2x) + 3x²(-3) + (-2x)(2x) + (-2x)(-3) + (1)(2x) + (1)(-3) = 6x³ - 9x² - 4x + 6x - 2x + 3, which simplifies to 6x³ - 9x² + 2.

III) P1 - P2

Last, we'll subtract P2 from P1: (3x² - 2x + 1) - (2x - 3) = 3x² - 2x + 1 - 2x + 3 = 3x² - 4x + 4. Again, like terms are combined.

In conclusion, eliminate terms wherever possible to simplify the algebra and check the answer to see if it is reasonable.