Multiply.
1) (3t^2 - 2t - 4) * (5t + 9)
Writing.
1) Explain why the product of a quadratic polynomial and a linear polynomial must be a cubic polynomial.
(3t^2 - 2t - 4) * (5t + 9)
Multiply each term in one set of parentheses by each term in the other set of parentheses.
3t^2(5t + 9) - 2t(5t + 9) - 4(5t + 9)
15t^3 + 27t^2 - 10t^2 - 18t - 20t - 36
combine like terms
15t^3 + (27t^2 - 10t^2) + (- 18t - 20t) - 36
15t^3 + 17t^2 -38t - 36
A quadratic has a power of 2 as its highest exponent, and a linear has a power of 1 as its highest exponent. When those highest exponents are multiplied together, they will result in a highest exponent of 2 + 1 = 3.
To multiply the given expressions (3t^2 - 2t - 4) and (5t + 9), we can use the distributive property.
First, distribute 3t^2 to both terms in the second expression:
(3t^2) * (5t + 9) = (15t^3 + 27t^2)
Next, distribute -2t to both terms in the second expression:
(-2t) * (5t + 9) = (-10t^2 - 18t)
Finally, distribute -4 to both terms in the second expression:
(-4) * (5t + 9) = (-20t - 36)
Combine the terms obtained in the previous steps to find the product:
(15t^3 + 27t^2) + (-10t^2 - 18t) + (-20t - 36)
Now, simplify the expression by combining like terms:
15t^3 + (27t^2 - 10t^2) + (-18t - 20t) - 36
This equals:
15t^3 + 17t^2 - 38t - 36
So, the product of the given quadratic polynomial (3t^2 - 2t - 4) and the linear polynomial (5t + 9) is a cubic polynomial, which is represented by the expression 15t^3 + 17t^2 - 38t - 36.
When multiplying a quadratic polynomial (degree 2) and a linear polynomial (degree 1), the resulting product will always have a degree equal to the sum of the degrees of the two polynomials. In this case, 2 + 1 = 3, which means the resulting product is a cubic polynomial (degree 3).