Simplifying Expressions & Expanding Brackets |
|
Simplifying Algebraic Expressions — Polypad
Junior Cycle • Algebra
Show that \(2x + 2y + 4 + 3x -3y +2\) simplifies to \(5x -y +6\)
Combining Like Terms — Algebra Tile Model
Step 1 — Expression as tiles:
xx
xxx
111
−x−x
−1
Step 2 — Like terms gathered:
x-tiles
xx
xxx
−x−x
constant tiles
111
−1
Step 3 — Simplified:
xxx
11
Simplified Expression: \(3x + 2\)
Step 1 — Expression as tiles:
x²
−x
x²x²
xxx
Step 2 — Like terms gathered:
\(x^2\)-tiles
x²
x²x²
x-tiles
xxx
−x
Step 3 — Simplified:
x²x²x²
xx
Simplified Expression: \(3x^2 + 2x\)
Step 1 — Expression as tiles:
aaa
−b
bbbb
aa
Step 2 — Like terms gathered:
a-tiles
aaa
aa
b-tiles
bbbb
−b
Step 3 — Simplified:
aaaaa
bbb
Simplified Expression: \(5a + 3b\)
Step 1 — Expression as tiles:
x²x²
1111
−x²
−1−1−1
−1−1−1
Step 2 — Like terms gathered:
\(x^2\)-tiles
x²x²
−x²
constant tiles
11
11
−1−1−1
−1−1−1
Step 3 — Simplified:
x²
−1−1
Simplified Expression: \(x^2 - 2\)
Step 1 — Expression as tiles:
y
b
yyy
11
−b−b
−1−1−1
−1−1
Step 2 — Like terms gathered:
y-tiles
y
yyy
b-tiles
b
−b−b
constant tiles
11
−1−1−1
−1−1
Step 3 — Simplified:
yyyy
−b
−1−1−1
Simplified Expression: \(4y - b - 3\)
Step 1 — Expression as tiles:
xx
a
xxx
111
11
−a−a
−a−a
−1−1
Step 2 — Like terms gathered:
x-tiles
xx
xxx
a-tiles
a
−a−a
−a−a
constant tiles
111
11
−1−1
Step 3 — Simplified:
xxx
xx
−a−a−a
111
Simplified Expression: \(5x - 3a + 3\)
Step 1 — Expression as tiles:
x²x²x²
x
−1−1
−1−1
xxx
xx
−x²−x²
11
11
Step 2 — Like terms gathered:
\(x^2\)-tiles
x²x²x²
−x²−x²
x-tiles
x
xx
xx
constant tiles
−1−1
−1−1
11
11
Step 3 — Simplified:
x²
xxx
xx
Simplified Expression: \(x^2 + 6x\)
Step 1 — Expression as tiles:
aaaa
111
111
−a−a
y
−y−y−y
−1
Step 2 — Like terms gathered:
a-tiles
aaaa
−a−a
y-tiles
y
−y−y
−y
constant tiles
111
111
−1
Step 3 — Simplified:
aa
−y−y
111
11
Simplified Expression: \(2a - 2y + 5\)
Simplifying Expressions Using Like Terms
Simplify each expression:
(i) \(4a+6b+6-2a+b-3\)
(ii) \(2x^{2}-3x-7-x^{2}-5x+3\)
(iii) \(6ab+2cd-ab+3cd\)
(iv) \(6x-xy+5x-7xy\)
(v) \(3a^{2}-2a-6a+4a^{2}-3\)
(vi) \(y^{2}-8y-3y^{2}+2y-3\).
(i) \(4a+6b+6-2a+b-3\)
(ii) \(2x^{2}-3x-7-x^{2}-5x+3\)
(iii) \(6ab+2cd-ab+3cd\)
(iv) \(6x-xy+5x-7xy\)
(v) \(3a^{2}-2a-6a+4a^{2}-3\)
(vi) \(y^{2}-8y-3y^{2}+2y-3\).
(i) \(4a+6b+6-2a+b-3\)
Like \(a\)-terms: \(4a-2a=2a\).
Like \(b\)-terms: \(6b+b=7b\).
Constants: \(6-3=3\).
Answer: \(2a+7b+3\).
(ii) \(2x^{2}-3x-7-x^{2}-5x+3\)
Like \(x^{2}\)-terms: \(2x^{2}-x^{2}=x^{2}\).
Like \(x\)-terms: \(-3x-5x=-8x\).
Constants: \(-7+3=-4\).
Answer: \(x^{2}-8x-4\).
(iii) \(6ab+2cd-ab+3cd\)
Like \(ab\)-terms: \(6ab-ab=5ab\).
Like \(cd\)-terms: \(2cd+3cd=5cd\).
Answer: \(5ab+5cd\).
(iv) \(6x-xy+5x-7xy\)
Like \(x\)-terms: \(6x+5x=11x\).
Like \(xy\)-terms: \(-xy-7xy=-8xy\).
Answer: \(11x-8xy\).
(v) \(3a^{2}-2a-6a+4a^{2}-3\)
Like \(a^{2}\)-terms: \(3a^{2}+4a^{2}=7a^{2}\).
Like \(a\)-terms: \(-2a-6a=-8a\).
Constants: \(-3\).
Answer: \(7a^{2}-8a-3\).
(vi) \(y^{2}-8y-3y^{2}+2y-3\)
Like \(y^{2}\)-terms: \(y^{2}-3y^{2}=-2y^{2}\).
Like \(y\)-terms: \(-8y+2y=-6y\).
Constants: \(-3\).
Answer: \(-2y^{2}-6y-3\).
Simplifying Expressions Quiz
Question 1 of 5
Score: 0 / 5
Multiplying Expressions by Integers — Algebra Tile Model
Expression as tiles:
\(2(\)
xxx
11
11
\()\)
Algebra (area model)
\(3x\)
\(4\)
\(2\)
Multiply by \(2\):
Algebra tiles
xxx
11
11
xxx
11
11
Algebra (area model)
\(3x\)
\(4\)
\(2\)
\(6x\)
\(8\)
Expanded Expression: \(2(3x + 4) = 6x + 8\)
Expression as tiles:
\(-1(\)
xxx
−1−1
−1−1
−1
\()\)
Algebra (area model)
\(3x\)
\(-5\)
\(-1\)
Multiply by \(-1\):
Algebra tiles
−x−x−x
11
111
Algebra (area model)
\(3x\)
\(-5\)
\(-1\)
\(-3x\)
\(5\)
Expanded Expression: \(-1(3x - 5) = -3x + 5\)
Expression as tiles:
\(2(\)
x²
−x−x
111
\()\)
Algebra (area model)
\(x^2\)
\(-2x\)
\(3\)
\(2\)
Multiply by \(2\):
Algebra tiles
x²
−x−x
111
x²
−x−x
111
Algebra (area model)
\(x^2\)
\(-2x\)
\(3\)
\(2\)
\(2x^2\)
\(-4x\)
\(6\)
Expanded Expression: \(2(x^2 - 2x + 3) = 2x^2 - 4x + 6\)
Expression as tiles:
\(3(\)
x²
−x−x
111
11
\()\)
Algebra (area model)
\(x^2\)
\(-2x\)
\(5\)
\(3\)
Multiply by \(3\):
Algebra tiles
x²
−x−x
111
11
x²
−x−x
111
11
x²
−x−x
111
11
Algebra (area model)
\(x^2\)
\(-2x\)
\(5\)
\(3\)
\(3x^2\)
\(-6x\)
\(15\)
Expanded Expression: \(3(x^2 - 2x + 5) = 3x^2 - 6x + 15\)
Expression as tiles:
\(4(\)
x²x²
x
−1−1−1
\()\)
Algebra (area model)
\(2x^2\)
\(x\)
\(-3\)
\(4\)
Multiply by \(4\):
Algebra tiles
x²x²
x
−1−1−1
x²x²
x
−1−1−1
x²x²
x
−1−1−1
x²x²
x
−1−1−1
Algebra (area model)
\(2x^2\)
\(x\)
\(-3\)
\(4\)
\(8x^2\)
\(4x\)
\(-12\)
Expanded Expression: \(4(2x^2 + x - 3) = 8x^2 + 4x - 12\)
Multiplying Terms by an Integer
Question 1 of 5
Score: 0 / 5
Multiplying Expressions by \(x\) — Algebra Tile Area Model
Expression as an area of tiles:
x
x
x
1
1
1
Algebra (area array)
\(2x\)
\(3\)
\(x\)
Multiply by \(x\): complete the area model and write the algebra.
x
x
x
1
1
1
x²
x²
x
x
x
Algebra (area array)
\(2x\)
\(3\)
\(x\)
\(2x^2\)
\(3x\)
Expanded Expression: \(x(2x + 3) = 2x^2 + 3x\)
Expression as an area of tiles:
x
x
x
x
x
−1
−1
Algebra (area array)
\(3x\)
\(-2\)
\(2x\)
Multiply by \(2x\): complete the area model and write the algebra.
x
x
x
x
x
−1
−1
x²
x²
x²
−x
−x
x²
x²
x²
−x
−x
Algebra (area array)
\(3x\)
\(-2\)
\(2x\)
\(6x^2\)
\(-4x\)
Expanded Expression: \(2x(3x - 2) = 6x^2 - 4x\)
Expression as an area of tiles:
−x
−x
−x
x
1
1
1
1
Algebra (area array)
\(x\)
\(4\)
\(-3x\)
Multiply by \(-3x\): complete the area model and write the algebra.
−x
−x
−x
x
1
1
1
1
−x²
−x
−x
−x
−x
−x²
−x
−x
−x
−x
−x²
−x
−x
−x
−x
Algebra (area array)
\(x\)
\(4\)
\(-3x\)
\(-3x^2\)
\(-12x\)
Expanded Expression: \(-3x(x + 4) = -3x^2 - 12x\)
Expanding — Polypad
Junior Cycle • Algebra
Show that the area of a rectangle with sides of \( (2x + 2) \) and \( (x + 3) \) is \( 2x^2 + 8x + 6 \)
Multiplying Binomials — Algebra Tile Area Model
Expression as an area of tiles:
x
1
1
x
1
1
1
x²
x
x
x
x
1
1
1
x
1
1
1
Algebra (area array)
\(x\)
\(3\)
\(x\)
\(2\)
Multiply: complete the area model and write the algebra.
x
1
1
x
1
1
1
x²
x
x
x
x
1
1
1
x
1
1
1
Algebra (area array)
\(x\)
\(3\)
\(x\)
\(x^2\)
\(3x\)
\(2\)
\(2x\)
\(6\)
Expanded Expression: \((x + 2)(x + 3) = x^2 + 5x + 6\)
Expression as an area of tiles:
x
x
1
x
−1
−1
−1
−1
x²
x
x
x
x
x²
x
x
x
x
x
1
1
1
1
Algebra (area array)
\(x\)
\(-4\)
\(2x\)
\(1\)
Multiply: complete the area model and write the algebra.
x
x
1
x
−1
−1
−1
−1
x²
−x
−x
−x
−x
x²
−x
−x
−x
−x
x
−1
−1
−1
−1
Algebra (area array)
\(x\)
\(-4\)
\(2x\)
\(2x^2\)
\(-8x\)
\(1\)
\(x\)
\(-4\)
Expanded Expression: \((2x + 1)(x - 4) = 2x^2 - 7x - 4\)
Expression as an area of tiles:
x
x
1
1
1
x
x
x
−1
−1
−1
−1
x²
x²
x²
x
x
x
x
x²
x²
x²
x
x
x
x
x
x
x
1
1
1
1
x
x
x
1
1
1
1
x
x
x
1
1
1
1
Algebra (area array)
\(3x\)
\(-4\)
\(2x\)
\(3\)
Multiply: complete the area model and write the algebra.
x
x
1
1
1
x
x
x
−1
−1
−1
−1
x²
x²
x²
−x
−x
−x
−x
x²
x²
x²
−x
−x
−x
−x
x
x
x
−1
−1
−1
−1
x
x
x
−1
−1
−1
−1
x
x
x
−1
−1
−1
−1
Algebra (area array)
\(3x\)
\(-4\)
\(2x\)
\(6x^2\)
\(-8x\)
\(3\)
\(9x\)
\(-12\)
Expanded Expression: \((2x + 3)(3x - 4) = 6x^2 + x - 12\)
Interactive Binomial Multiplier — Algebra Tile Area Model
Expand
Step 1 — Arrange the factors as side and top of a rectangle:
Algebra (area array)
Step 2 — Fill the rectangle and expand the product:
Algebra (area array)
Expanded expression:
Multiplying Binomials (Area Model)
Multiply each of the following and simplify your answer:
(i) \((x+4)(x-2)\)
(ii) \((x-3)(x+5)\).
(i) \((x+4)(x-2)\)
(ii) \((x-3)(x+5)\).
(i) \((x+4)(x-2)\)
Area model:
\(x\)
\(-2\)
\(x\)
\(4\)
\(x^{2}\)
\(-2x\)
\(4x\)
\(-8\)
Add all the areas: \(x^{2}-2x+4x-8=x^{2}+2x-8\).
Answer: \(x^{2}+2x-8\).
(ii) \((x-3)(x+5)\)
Area model:
\(x\)
\(5\)
\(x\)
\(-3\)
\(x^{2}\)
\(5x\)
\(-3x\)
\(-15\)
Add all the areas: \(x^{2}+5x-3x-15=x^{2}+2x-15\).
Answer: \(x^{2}+2x-15\).
Multiplying Two Expressions Quiz
Question 1 of 5
Score: 0 / 5
Expanding Brackets (Separation Method)
Remove the brackets and simplify each of these:
(i) \((2x-3)(3x-4)\)
(ii) \((2x-4)(x^{2}-3x+5)\).
(i) \((2x-3)(3x-4)\)
(ii) \((2x-4)(x^{2}-3x+5)\).
(i) \((2x-3)(3x-4)\)
Separate: \(2x(3x-4)-3(3x-4)\).
First part: \(2x(3x-4)=6x^{2}-8x\).
Second part: \(-3(3x-4)=-9x+12\).
Combine like terms: \(6x^{2}-8x-9x+12=6x^{2}-17x+12\).
Answer: \(6x^{2}-17x+12\).
(ii) \((2x-4)(x^{2}-3x+5)\)
Separate: \(2x(x^{2}-3x+5)-4(x^{2}-3x+5)\).
First part: \(2x(x^{2}-3x+5)=2x^{3}-6x^{2}+10x\).
Second part: \(-4(x^{2}-3x+5)=-4x^{2}+12x-20\).
Combine like terms:
\(2x^{3}-(6x^{2}+4x^{2})+(10x+12x)-20 =2x^{3}-10x^{2}+22x-20\).
\(2x^{3}-(6x^{2}+4x^{2})+(10x+12x)-20 =2x^{3}-10x^{2}+22x-20\).
Answer: \(2x^{3}-10x^{2}+22x-20\).
