Answer :
To find the product of the polynomials
[tex]$$ (3x^2 - 5x + 3)(3x - 2), $$[/tex]
we multiply each term in the first polynomial by each term in the second polynomial.
1. Multiply the [tex]$x^2$[/tex] term:
[tex]$$ 3x^2 \cdot 3x = 9x^3, $$[/tex]
[tex]$$ 3x^2 \cdot (-2) = -6x^2. $$[/tex]
2. Multiply the [tex]$x$[/tex] term:
[tex]$$ -5x \cdot 3x = -15x^2, $$[/tex]
[tex]$$ -5x \cdot (-2) = 10x. $$[/tex]
3. Multiply the constant term:
[tex]$$ 3 \cdot 3x = 9x, $$[/tex]
[tex]$$ 3 \cdot (-2) = -6. $$[/tex]
Now, list all the terms:
[tex]$$ 9x^3, \quad -6x^2, \quad -15x^2, \quad 10x, \quad 9x, \quad -6. $$[/tex]
Next, combine like terms:
- The [tex]$x^3$[/tex] term is: [tex]$$ 9x^3. $$[/tex]
- The [tex]$x^2$[/tex] terms are: [tex]$$ -6x^2 - 15x^2 = -21x^2. $$[/tex]
- The [tex]$x$[/tex] terms are: [tex]$$ 10x + 9x = 19x. $$[/tex]
- The constant term is: [tex]$$ -6. $$[/tex]
Thus, the product is:
[tex]$$ 9x^3 - 21x^2 + 19x - 6. $$[/tex]
[tex]$$ (3x^2 - 5x + 3)(3x - 2), $$[/tex]
we multiply each term in the first polynomial by each term in the second polynomial.
1. Multiply the [tex]$x^2$[/tex] term:
[tex]$$ 3x^2 \cdot 3x = 9x^3, $$[/tex]
[tex]$$ 3x^2 \cdot (-2) = -6x^2. $$[/tex]
2. Multiply the [tex]$x$[/tex] term:
[tex]$$ -5x \cdot 3x = -15x^2, $$[/tex]
[tex]$$ -5x \cdot (-2) = 10x. $$[/tex]
3. Multiply the constant term:
[tex]$$ 3 \cdot 3x = 9x, $$[/tex]
[tex]$$ 3 \cdot (-2) = -6. $$[/tex]
Now, list all the terms:
[tex]$$ 9x^3, \quad -6x^2, \quad -15x^2, \quad 10x, \quad 9x, \quad -6. $$[/tex]
Next, combine like terms:
- The [tex]$x^3$[/tex] term is: [tex]$$ 9x^3. $$[/tex]
- The [tex]$x^2$[/tex] terms are: [tex]$$ -6x^2 - 15x^2 = -21x^2. $$[/tex]
- The [tex]$x$[/tex] terms are: [tex]$$ 10x + 9x = 19x. $$[/tex]
- The constant term is: [tex]$$ -6. $$[/tex]
Thus, the product is:
[tex]$$ 9x^3 - 21x^2 + 19x - 6. $$[/tex]
