Multiply the polynomials (3x^2+5x+9)(6x-8)

18x^3 +6x^2 +14x -72

Step-by-step explanation:

The highest degrees of the factors are 2 and 1, so the highest-degree term in the product will be of degree 2+1 = 3.

The product's degree 3 term is: 3x^2·6x = 18x^3.

The product's degree 2 term is: 3x^2·(-8) + 5x·6x = (-24+30)x^2 = 6x^2

The product's degree 1 term is: 5x·(-8) +9·6x = (-40 +54)x = 14x

The product's constant term is 9·(-8) = -72

The result is ...

... 18x^3 +6x^2 +14x -72

Comment on the above procedure

With some practice, you can learn to identify the contributors to each product term and do the math in your head. If you write the coefficients in an array, the pattern of products can be more obvious. Here, the coefficients would be ...

... 3   5   9

... 0   6   -8

If these were two 2nd-degree polynomials, the highest (4th) degree product term would have the coefficient 3 × 0, the product of the two left-most coefficients.

The 3rd-degree product term is the sum of the coefficient products taken in an X pattern: 3×6 + 5×0.

The 2nd-degree product term is the sum of terms of a wider X plus the product of the two middle coefficients: 3×-8 + 9×0 + 5×6.

As we shift our focus to lower-degree product terms, the X gets narrower again. The 1st-degree term is the sum: 5×-8 + 9×6. You will note that this is symmetrical with the X used to compute the 3rd-degree term.

Finally, the 0-degree term (constant) in the product is the product of the two right-most coefficients: 9×-8.

Distributive Property

The above is an alternative method to the use of the distributive property, which also gives you the product of every term in one factor with every term in the other factor. After doing that math, you have to be careful to collect terms properly.

(3x^2 +5x +9)(6x -8) = (3x^2 +5x +9)(6x) + (3x^2 +5x +9)(-8)

... = (18x^3 +30x^2 +54x) + (-24x^2 -40x -72)

... = 18x^3 +(30 -24)x^2 +(54 -40)x -72

... = 18x^3 +6x^2 +14x -72

18 +6 +14x -72

Step-by-step explanation: