Question

Multiply: $\left( {2{x^2} - 3x + 5} \right)$ by $\left( {5x + 2} \right)$
(A) $10{x^3} - 15{x^2} + 19x + 10$
(B) $10{x^3} - 11{x^2} + 19x + 10$
(C) $10{x^3} + 11{x^2} + 19x + 10$
(D) None of the above

Answer 1

Hint: To multiply polynomials, first multiply each term in one polynomial using distributive law, i.e., $a \times \left( {b + c} \right) = \left( {a \times b} \right) + \left( {a \times c} \right)$ .Then , simplify the resulting polynomial by adding or subtracting the like terms.

Complete step-by-step answer:
We have to multiply $\left( {2{x^2} - 3x + 5} \right)$ by $\left( {5x + 2} \right)$, so it can be written as:
\[\left( {2{x^2} - 3x + 5} \right) \times \left( {5x + 2} \right)\]
Now, use distributive law, i.e., $a \times \left( {b + c} \right) = \left( {a \times b} \right) + \left( {a \times c} \right)$ and separate the second polynomial.
$ = \left( {2{x^2} - 3x + 5} \right) \times 5x + \left( {2{x^2} - 3x + 5} \right) \times 2$
Multiply the monomials from the second polynomial with each term of the first polynomial.
$ = \left( {2{x^2} \times 5x - 3x \times 5x + 5 \times 5x} \right) + \left( {2{x^2} \times 2 - 3x \times 2 + 5 \times 2} \right)$
$ = \left( {10{x^3} - 15{x^2} + 25x} \right) + \left( {4{x^2} - 6x + 10} \right)$
Simplify the resultant polynomial.
$ = 10{x^3} - 15{x^2} + 4{x^2} + 25x - 6x + 10$
$ = 10{x^3} - 11{x^2} + 19x + 10$

Hence, option (B) is the correct answer.

Note: Distributive law of multiplication is used twice when 2 polynomials are multiplied.
Look for the like terms and combine them. This may reduce the expected number of terms in the product.
Preferably, write the terms in the decreasing order of their exponent.
Be careful with the signs, when you open the brackets.
The resulting degree after multiplying two polynomials will be higher than the degree of the individual polynomials.