Multiply the binomials: (2x+5) and (4x−3); (y−8) and (3y−4); (2.5l − 0.5m) and (2.5l + 0.5m); (a+3b) and…

Question

Multiply the binomials: (2x+5) and (4x−3); (y−8) and (3y−4); (2.5l − 0.5m) and (2.5l + 0.5m); (a+3b) and (x+5); (2pq+3q2) and (3pq-2q2).

ScanSolve AI Tutor · Accepted Answer

Here are the solutions for the 1st question:

1. Multiply the binomials.

Use the distributive property (FOIL method).
$$ (2x+5)(4x-3) = 2x(4x) + 2x(-3) + 5(4x) + 5(-3) $$
Perform the multiplications.
$$ = 8x^2 - 6x + 20x - 15 $$
Combine like terms.
$$ = \boxed{8x^2 + 14x - 15} $$
- 1 (ii): $(y-8)$ and $(3y-4)$
Use the distributive property (FOIL method).
$$ (y-8)(3y-4) = y(3y) + y(-4) - 8(3y) - 8(-4) $$
Perform the multiplications.
$$ = 3y^2 - 4y - 24y + 32 $$
Combine like terms.
$$ = \boxed{3y^2 - 28y + 32} $$
- 1 (iii): $(2.5l - 0.5m)$ and $(2.5l + 0.5m)$
Recognize this as a difference of squares
$(A-B)(A+B) = A^2 - B^2$ . Here $A=2.5l$ and $B=0.5m$ . $(2.5l - 0.5m)(2.5l + 0.5m) = (2.5l)^2 - (0.5m)^2$
Square each term.
$$ = (2.5)^2 l^2 - (0.5)^2 m^2 $$ $$ = \boxed{6.25l^2 - 0.25m^2} $$
- 1 (iv): $(a+3b)$ and $(x+5)$
Use the distributive property (FOIL method).
$$ (a+3b)(x+5) = a(x) + a(5) + 3b(x) + 3b(5) $$
Perform the multiplications.
$$ = \boxed{ax + 5a + 3bx + 15b} $$
- 1 (v): $(2pq+3q^2)$ and $(3pq-2q^2)$

$$
(2pq+3q^2)(3pq-2q^2) = 2pq(3pq) + 2pq(-2q^2) + 3q^2(3pq) + 3q^2(-2q^2)
$$

Perform the multiplications.
$$ = 6p^2q^2 - 4pq^3 + 9pq^3 - 6q^4 $$
Combine like terms.
$$ = \boxed{6p^2q^2 + 5pq^3 - 6q^4} $$
- 1 (vi): $(\frac{3}{4}a^2+3b^2)$ and $4(a^2-\frac{2}{3}b^2)$

$$
4(a^2-\frac{2}{3}b^2) = 4a^2 - 4 \times \frac{2}{3}b^2 = 4a^2 - \frac{8}{3}b^2
$$

Multiply the two binomials
$(\frac{3}{4}a^2+3b^2)(4a^2-\frac{8}{3}b^2)$ . $= \frac{3}{4}a^2(4a^2) + \frac{3}{4}a^2(-\frac{8}{3}b^2) + 3b^2(4a^2) + 3b^2(-\frac{8}{3}b^2)$

$$
= 3a^4 - \frac{24}{12}a^2b^2 + 12a^2b^2 - \frac{24}{3}b^4
$$

$$
= 3a^4 - 2a^2b^2 + 12a^2b^2 - 8b^4
$$
$$
= \boxed{3a^4 + 10a^2b^2 - 8b^4}
$$

Multiply the binomials: (2x+5) and (4x-3); (y-8) and (3y-4); (2.5l - 0.5m) and (2.5l + 0.5m); (a+3b) and (x+5); (2pq+3q2) and (3pq-2q2).