Question

Multiply the polynomials using the special product formulas. Express your answer as a single polynomial in standard form. $$ (x+5)^{2} $$

Short answer

x^{2} + 10x + 25

Step-by-step solution

Identify the Formula

The given expression \( (x+5)^{2} \) can be multiplied using the square of a binomial formula, which is \( (a+b)^{2} = a^{2} + 2ab + b^{2} \).

Apply the Formula

In this case, \( a = x \) and \( b = 5 \). Substitute these values into the formula: \( (x+5)^{2} = x^{2} + 2(x)(5) + 5^{2} \).

Simplify Each Term

Calculate each term separately:- \( x^{2} \) is already simplified.- \[ 2(x)(5) = 10x \].- \( 5^{2} = 25 \).

Combine All Terms

Now combine all the simplified terms to get the final polynomial: \[ x^{2} + 10x + 25 \].

Write in Standard Form

The final expression \[ x^{2} + 10x + 25 \] is already in standard form, with terms ordered by decreasing exponent.

Key concepts

special product formulas

Special product formulas are shortcuts that help simplify the multiplication of polynomials. They save time and reduce errors. One of these formulas is the square of a binomial formula, which we often use when we square expressions like \((x + 5)^2\).
The key special product formulas include:

• Square of a Binomial: \[ (a + b)^2 = a^2 + 2ab + b^2 \] or \[ (a - b)^2 = a^2 - 2ab + b^2 \]
• Product of a Sum and a Difference: \[ (a + b)(a - b) = a^2 - b^2 \]
• Sum or Difference of Cubes: \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \] and \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

Knowing these formulas can make polynomial multiplication much quicker and easier.

square of a binomial

The square of a binomial is a specific special product formula. When you're squaring a binomial like \((x + 5)^2\), you apply the formula \[ (a + b)^2 = a^2 + 2ab + b^2 \] to simplify it.

Here's a quick breakdown using our example \( (x + 5)^2 \):

• Identify a and b: In this example, \(a = x\) and \(b = 5\).
• Apply the formula: Substitute \(a\) and \(b\) into the formula to get \[ (x + 5)^2 = x^2 + 2(x)(5) + 5^2 \]
• Simplify each term:
  • \(x^2\) stays as it is.
  • \[2(x)(5) = 10x \]
  • \(5^2 = 25\)
• Combine the terms to get \[ x^2 + 10x + 25 \]

This formula helps to expand binomials quickly without manually multiplying each term.

polynomial simplification

Polynomial simplification involves combining like terms and reducing the expression to its simplest form. After expanding a binomial using the square of a binomial formula, the next step is simplifying.

Here's how you simplify a polynomial:

• Combine like terms: Group together terms that have the same variables raised to the same power.
• Perform arithmetic operations: Carry out any addition, subtraction, multiplication, or division needed.

For example, after expanding \( (x + 5)^2\) to \[ x^2 + 10x + 25 \], it's already simplified. No like terms need combining, and all calculations are complete.

standard form of polynomials

The standard form of a polynomial is a way of writing it so that the terms are arranged in descending order of their exponents. This makes the polynomial easier to read and understand.
When writing a polynomial in standard form, remember to:

• Order terms by their exponents, highest to lowest. For instance, \[ x^2 + 10x + 25 \] is in standard form.
• Combine like terms, if there are any. This ensures the polynomial is entirely simplified.
• Write with positive exponent order. Ensure that every term is fully simplified and all the like terms are combined.

Standard form helps in understanding the structure and degree of the polynomial more clearly.