Q.211

Question

In the following exercises, multiply using a) the Distributive Property; b) the Vertical Method

$(2 a^{2} - 5 a + 10) (3 a^{2} + a - 5)$

Step-by-Step Solution

Verified

Answer

The product of the given polynomials is $6 a^{4} - 13 a^{3} + 15 a^{2} + 35 a - 50$

1Step 1. Given Information

The given polynomials are $(2 a^{2} - 5 a + 10) (3 a^{2} + a - 5)$

The product is to be determined using a) the Distributive Property; b) the Vertical Method

2Part (a) Step 1. Using distributive property

According to distributive property:

$(2 a^{2} - 5 a + 10) (3 a^{2} + a - 5) = 2 a^{2} (3 a^{2} + a - 5) - 5 a (3 a^{2} + a - 5) + 10 (3 a^{2} + a - 5) (2 a^{2} - 5 a + 10) (3 a^{2} + a - 5) = 6 a^{4} + 2 a^{3} - 10 a^{2} - 15 a^{3} - 5 a^{2} + 25 a + 30 a^{2} + 10 a - 50 (2 a^{2} - 5 a + 10) (3 a^{2} + a - 5) = 6 a^{4} - 13 a^{3} + 15 a^{2} + 35 a - 50$

3Part (a) Step 2. Conclusion

The product of the polynomials is $6 a^{4} - 13 a^{3} + 15 a^{2} + 35 a - 50$

4Part (b) Step 1. Using Vertical Method

Multiplying using vertical method:

$\begin{array}{rrrr} 3 a^{2} + & a - & 5 \\ \times & 2 a^{2} - & 5 a + & 10 \\ 30 a^{2} + & 10 a - & 50 \\ - 15 a^{3} - & 5 a^{2} + & 25 a \\ 6 a^{4} + 2 a^{3} - & 10 a^{2} \\ 6 a^{4} - 13 a^{3} & + 15 a^{2} + & 35 a - & 50 \end{array}$

5Part (b) Step 2. Conclusion

The product of the polynomials is $6 a^{4} - 13 a^{3} + 15 a^{2} + 35 a - 50$

Q.210

Q.212

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