Monomial Multiplication

1. Multiply: $6x \times 4y$

2. Multiply: $5a \times (-3a^2b)$

3. Multiply: $3x^2y \times 4xy^2$

4. Multiply: $-2ab \times 7a^2b^3$

Monomial by Polynomial

5. Find the product: $4x(2x + 5y)$

6. Find the product: $a^2(3ab – 4c)$

Binomial Multiplication

7. Multiply: $(x – 5)(2x + 3)$

8. Multiply: $(a + b)(3a – 2b)$

Simplification

9. Simplify: $(p + q)(2p – 3q) – (p – q)q$

10. Simplify: $3t(t – 2) + 2t(2t + 5) – t^2$

Monomial Multiplication

1. $6x \times 4y = 24xy$ Multiply the coefficients ($6 \times 4$) and then append the variables ($x$ and $y$). Since the variables are different, they are simply written together.

2. $5a \times (-3a^2b) = -15a^3b$ Multiply the numbers ($5 \times -3 = -15$). For the $a$ terms, add the exponents ($1 + 2 = 3$) to get $a^3$. The $b$ remains as is.

3. $3x^2y \times 4xy^2 = 12x^3y^3$ Multiply coefficients ($3 \times 4 = 12$). Add the exponents for $x$ ($2 + 1 = 3$) and the exponents for $y$ ($1 + 2 = 3$).

4. $-2ab \times 7a^2b^3 = -14a^3b^4$ Multiply the coefficients ($-2 \times 7 = -14$). Combine variables by adding their powers: $a^{(1+2)}$ and $b^{(1+3)}$.

Monomial by Polynomial

5. $4x(2x + 5y) = 8x^2 + 20xy$ Use the Distributive Property. Multiply $4x$ by the first term ($2x$) to get $8x^2$, then multiply $4x$ by the second term ($5y$) to get $20xy$.

6. $a^2(3ab – 4c) = 3a^3b – 4a^2c$ Distribute $a^2$ to both terms inside the parentheses. $a^2 \times 3ab$ becomes $3a^3b$, and $a^2 \times -4c$ becomes $-4a^2c$.

Binomial Multiplication

7. $(x – 5)(2x + 3) = 2x^2 – 7x – 15$ Use the FOIL method (First, Outer, Inner, Last):

   • First: $x \times 2x = 2x^2$

   • Outer: $x \times 3 = 3x$

   • Inner: $-5 \times 2x = -10x$

   • Last: $-5 \times 3 = -15$

   • Combine like terms ($3x – 10x = -7x$).

8. $(a + b)(3a – 2b) = 3a^2 + ab – 2b^2$ Apply FOIL:

   • First: $a \times 3a = 3a^2$

   • Outer: $a \times -2b = -2ab$

   • Inner: $b \times 3a = 3ab$

   • Last: $b \times -2b = -2b^2$

   • Combine like terms ($-2ab + 3ab = 1ab$ or just $ab$).

Simplification

9. $(p + q)(2p – 3q) – (p – q)q = 2p^2 – 2pq – 2q^2$

   • Expand the first part using FOIL: $2p^2 – 3pq + 2pq – 3q^2$, which simplifies to $2p^2 – pq – 3q^2$.

   • Expand the second part: $q(p – q) = pq – q^2$.

   • Subtract the second part from the first: $(2p^2 – pq – 3q^2) – (pq – q^2)$.

   • Distribute the negative sign: $2p^2 – pq – 3q^2 – pq + q^2$.

   • Combine like terms: $2p^2 – 2pq – 2q^2$.

10. $3t(t – 2) + 2t(2t + 5) – t^2 = 6t^2 + 4t$

    • Distribute $3t$: $3t^2 – 6t$.

    • Distribute $2t$: $4t^2 + 10t$.

    • Combine all terms: $(3t^2 + 4t^2 – t^2) + (-6t + 10t)$.

    • Result: $6t^2 + 4t$.