To determine if the expression [tex]\( 7x^2 + 1 \)[/tex] is a quadratic binomial, let’s break down the terms and definitions we need:
1. Quadratic: An expression is considered quadratic if it includes a term with [tex]\( x^2 \)[/tex]. A general form for a quadratic expression is [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants and [tex]\( a \neq 0 \)[/tex].
2. Binomial: A binomial is an algebraic expression that contains exactly two terms.
Let's examine the expression [tex]\( 7x^2 + 1 \)[/tex]:
- First term: [tex]\( 7x^2 \)[/tex]. This term is quadratic because it has [tex]\( x \)[/tex] raised to the power of 2.
- Second term: [tex]\( 1 \)[/tex]. This is a constant term (a number without any variables).
Checking against the definitions:
1. Quadratic Check: The expression [tex]\( 7x^2 + 1 \)[/tex] fits the quadratic form [tex]\( ax^2 + bx + c \)[/tex]:
- Here, [tex]\( a = 7 \)[/tex] (coefficient of [tex]\( x^2 \)[/tex]),
- [tex]\( b = 0 \)[/tex] (no [tex]\( x \)[/tex] term is present, thus [tex]\( b = 0 \)[/tex]),
- [tex]\( c = 1 \)[/tex] (constant term).
Since [tex]\( a \neq 0 \)[/tex] (in this case, [tex]\( a = 7 \)[/tex]), the expression is indeed quadratic.
2. Binomial Check: The expression [tex]\( 7x^2 + 1 \)[/tex] has exactly two terms: [tex]\( 7x^2 \)[/tex] and [tex]\( 1 \)[/tex]. Therefore, it qualifies as a binomial.
Since [tex]\( 7x^2 + 1 \)[/tex] fulfills the criteria of being both quadratic and a binomial, we can conclude that the expression [tex]\( 7x^2 + 1 \)[/tex] is a quadratic binomial.
Thus, the answer to the question is:
True
