Which expressions are perfect square trinomials? Check all that apply
Perfect square trinomials are a fundamental concept in algebra, often used to simplify quadratic expressions and factorize polynomials. Recognizing a perfect square trinomial is crucial for understanding various algebraic operations. In this article, we will discuss the characteristics of perfect square trinomials and identify which expressions among a given set qualify as perfect square trinomials.
A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. It has the general form (a + b)^2 or (a – b)^2, where ‘a’ and ‘b’ are real numbers. The key feature of a perfect square trinomial is that its first and last terms are perfect squares, and the middle term is twice the product of the square roots of the first and last terms.
Let’s analyze the given expressions to determine which ones are perfect square trinomials:
1. x^2 + 6x + 9
2. 4x^2 – 8x + 4
3. 9x^2 – 6x + 1
4. x^2 + 4x + 4
5. 25x^2 – 10x + 1
To check if an expression is a perfect square trinomial, we need to ensure that:
– The first and last terms are perfect squares.
– The middle term is twice the product of the square roots of the first and last terms.
Let’s examine each expression:
1. x^2 + 6x + 9
The first term, x^2, is a perfect square. The last term, 9, is also a perfect square (3^2). The middle term, 6x, is twice the product of the square roots of the first and last terms (2 x 3). Therefore, this expression is a perfect square trinomial.
2. 4x^2 – 8x + 4
The first term, 4x^2, is a perfect square (2x)^2. The last term, 4, is also a perfect square (2^2). The middle term, -8x, is twice the product of the square roots of the first and last terms (2 2x 2). Hence, this expression is a perfect square trinomial.
3. 9x^2 – 6x + 1
The first term, 9x^2, is a perfect square (3x)^2. The last term, 1, is a perfect square (1^2). The middle term, -6x, is twice the product of the square roots of the first and last terms (2 3x 1). Therefore, this expression is a perfect square trinomial.
4. x^2 + 4x + 4
The first term, x^2, is a perfect square. The last term, 4, is a perfect square (2^2). The middle term, 4x, is twice the product of the square roots of the first and last terms (2 x 2). Hence, this expression is a perfect square trinomial.
5. 25x^2 – 10x + 1
The first term, 25x^2, is a perfect square (5x)^2. The last term, 1, is a perfect square (1^2). The middle term, -10x, is twice the product of the square roots of the first and last terms (2 5x 1). Therefore, this expression is a perfect square trinomial.
In conclusion, all the given expressions are perfect square trinomials. Recognizing these expressions is essential for various algebraic operations and simplifications.
