Question

Remove the brackets from the following and simplify the result: (a) \((a+3)(a-5)\) (b) \((2 b+6)(b-7)\) (c) \((3 c+4)(2 c-1)\) (d) \((5 d+8)(-2 d-1)\)

Short answer

Question: Simplify the following expressions and write the final expressions: a) (a+3)(a-5) b) (2b+6)(b-7) c) (3c+4)(2c-1) d) (5d+8)(-2d-1) Answer: a) a^2 - 2a - 15 b) 2b^2 - 8b - 42 c) 6c^2 + 5c - 4 d) -10d^2 - 21d - 8

Step-by-step solution

Simplifying expression (a)

Distribute the terms, according to the distributive property: \((a+3)(a-5) = a(a-5) + 3(a-5)\) Apply the distributive property: \(a(a) - a(5) + 3(a) - 3(5)\) Combine like terms and simplify: \(a^2 - 5a + 3a - 15\) Final expression: \(a^2 - 2a - 15\)

Simplifying expression (b)

Distribute the terms, according to the distributive property: \((2b+6)(b-7) = 2b(b-7) + 6(b-7)\) Apply the distributive property: \(2b(b) - 2b(7) + 6(b) - 6(7)\) Combine like terms and simplify: \(2b^2 - 14b + 6b - 42\) Final expression: \(2b^2 - 8b - 42\)

Simplifying expression (c)

Distribute the terms, according to the distributive property: \((3c+4)(2c-1) = 3c(2c-1) + 4(2c-1)\) Apply the distributive property: \(3c(2c) - 3c(1) + 4(2c) - 4(1)\) Combine like terms and simplify: \(6c^2 - 3c + 8c - 4\) Final expression: \(6c^2 + 5c - 4\)

Simplifying expression (d)

Distribute the terms, according to the distributive property: \((5d+8)(-2d-1) = 5d(-2d-1) + 8(-2d-1)\) Apply the distributive property: \(5d(-2d) - 5d(1) - 8(2d) - 8(1)\) Combine like terms and simplify: \(-10d^2 - 5d - 16d - 8\) Final expression: \(-10d^2 - 21d - 8\)

Key concepts

Distributive Property

The distributive property is a foundational concept in algebra that allows you to remove parentheses in expressions. It states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products. In simpler terms, when you have an expression like \(a(b + c)\), you can distribute the \(a\) to both \(b\) and \(c\), which means: \(a \cdot b + a \cdot c\).
This principle is crucial for simplifying expressions like those in our exercise. For example, in \( (a+3)(a-5) \), we apply the distributive property as follows:\[a(a-5) + 3(a-5)\]
Here, \(a\) is distributed over \(a-5\) and \(3\) is also distributed over \(a-5\). This results in smaller terms that can be further simplified.

Combining Like Terms

After distributing, the next step is to simplify the expression by combining like terms. Like terms are terms that have the same variable raised to the same power. For example, in the expression \(a^2 - 5a + 3a - 15\), \(-5a\) and \(3a\) are like terms since they both involve \(a\).

• You simply add or subtract the coefficients of these like terms. For instance, \(-5a + 3a = -2a\).
• Be sure to keep track of the signs (positive or negative), as they greatly affect the outcome.

Combining like terms effectively reduces the complexity of the expression, resulting in the final simplified form, such as \(a^2 - 2a - 15\).

Binomial Expansion

Binomial expansion involves multiplying two binomials to get a quadratic expression. A binomial has two terms, such as \(a + 3\). When expanding an expression like \( (a+3)(a-5) \), each term in the first binomial is multiplied by each term in the second binomial.
This process follows the distributive property:

• The term \(a\) in \(a + 3\) multiplies both \(a\) and \(-5\).
• The term \(3\) also multiplies both \(a\) and \(-5\).

After distributing, your expanded equation includes all the resulting terms, like \(a^2 - 5a + 3a - 15\), which can then be simplified.

Quadratic Expressions

A quadratic expression is a polynomial where the highest exponent of the variable is 2. It typically looks like \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. This form results commonly from the expansion of binomials.
For example, after expanding and simplifying \( (a+3)(a-5) \), the final quadratic expression is \(a^2 - 2a - 15\). Key features of quadratic expressions include:

• The quadratic term \(a^2\), which indicates the degree of the polynomial.
• Linear terms like \(-2a\), which help describe the slope or rate of change.
• Constant terms such as \(-15\), which shift the expression up or down the y-axis.

Understanding these components helps in graphing the expression and solving quadratic equations.