To solve the given problems, we need to expand each expression using the distributive property (also known as the FOIL method for binomials). Let's go through each problem step by step.
---
1. \((x + 2)(x - 3)\)
Using the distributive property:
\[
(x + 2)(x - 3) = x(x - 3) + 2(x - 3)
\]
\[
= x^2 - 3x + 2x - 6
\]
\[
= x^2 - x - 6
\]
Answer:
\[
\boxed{x^2 - x - 6}
\]
---
2. \((2y - 1)(y - 2)\)
Using the distributive property:
\[
(2y - 1)(y - 2) = 2y(y - 2) - 1(y - 2)
\]
\[
= 2y^2 - 4y - y + 2
\]
\[
= 2y^2 - 5y + 2
\]
Answer:
\[
\boxed{2y^2 - 5y + 2}
\]
---
3. \((x - 1)(x - 1)\)
This is a special case of squaring a binomial:
\[
(x - 1)(x - 1) = (x - 1)^2
\]
Using the formula \((a - b)^2 = a^2 - 2ab + b^2\):
\[
(x - 1)^2 = x^2 - 2(x)(1) + 1^2
\]
\[
= x^2 - 2x + 1
\]
Answer:
\[
\boxed{x^2 - 2x + 1}
\]
---
4. \((x - 2)(x + 2)\)
This is a difference of squares:
\[
(x - 2)(x + 2) = x^2 - 2^2
\]
\[
= x^2 - 4
\]
Answer:
\[
\boxed{x^2 - 4}
\]
---
5. \((2x - 5)(x - 1)\)
Using the distributive property:
\[
(2x - 5)(x - 1) = 2x(x - 1) - 5(x - 1)
\]
\[
= 2x^2 - 2x - 5x + 5
\]
\[
= 2x^2 - 7x + 5
\]
Answer:
\[
\boxed{2x^2 - 7x + 5}
\]
---
6. \((3 - y)(2 - y)\)
Using the distributive property:
\[
(3 - y)(2 - y) = 3(2 - y) - y(2 - y)
\]
\[
= 6 - 3y - 2y + y^2
\]
\[
= y^2 - 5y + 6
\]
Answer:
\[
\boxed{y^2 - 5y + 6}
\]
---
7. \((x - 2)(y - 3)\)
Using the distributive property:
\[
(x - 2)(y - 3) = x(y - 3) - 2(y - 3)
\]
\[
= xy - 3x - 2y + 6
\]
Answer:
\[
\boxed{xy - 3x - 2y + 6}
\]
---
8. \((2x - 3)(3x - 1)\)
Using the distributive property:
\[
(2x - 3)(3x - 1) = 2x(3x - 1) - 3(3x - 1)
\]
\[
= 6x^2 - 2x - 9x + 3
\]
\[
= 6x^2 - 11x + 3
\]
Answer:
\[
\boxed{6x^2 - 11x + 3}
\]
---
9. \((y - 2x)(y - x)\)
Using the distributive property:
\[
(y - 2x)(y - x) = y(y - x) - 2x(y - x)
\]
\[
= y^2 - xy - 2xy + 2x^2
\]
\[
= y^2 - 3xy + 2x^2
\]
Answer:
\[
\boxed{y^2 - 3xy + 2x^2}
\]
---
10. \((x - 1)(4 - x)\)
Using the distributive property:
\[
(x - 1)(4 - x) = x(4 - x) - 1(4 - x)
\]
\[
= 4x - x^2 - 4 + x
\]
\[
= -x^2 + 5x - 4
\]
Answer:
\[
\boxed{-x^2 + 5x - 4}
\]
---
Final Answers:
1. \(\boxed{x^2 - x - 6}\)
2. \(\boxed{2y^2 - 5y + 2}\)
3. \(\boxed{x^2 - 2x + 1}\)
4. \(\boxed{x^2 - 4}\)
5. \(\boxed{2x^2 - 7x + 5}\)
6. \(\boxed{y^2 - 5y + 6}\)
7. \(\boxed{xy - 3x - 2y + 6}\)
8. \(\boxed{6x^2 - 11x + 3}\)
9. \(\boxed{y^2 - 3xy + 2x^2}\)
10. \(\boxed{-x^2 + 5x - 4}\)
Parent Tip: Review the logic above to help your child master the concept of multiplying polynomials by monomials worksheet.
