Respuesta :
The number of unit tiles needed to be added to the expression [tex]x^2 + 4x + 3[/tex] for making it perfect square trinomial is: Option A: 1
What is a perfect square polynomial?
If a polynomial p(x) can be written as:
[tex]p(x) = [f(x)]^2[/tex]
where f(x) is also a polynomial, then p(x) is called as perfect square polynomial.
What are terms in polynomials?
Terms are added or subtracted to make a polynomial. They're composed of variables and constants all in multiplication.
Example: [tex]x^2 + 3x + 4[/tex] is a polynomial consisting 3 terms as [tex]x^2 ,3x , \rm and \: 4[/tex]
- If there is one term, the polynomial will be called monomial.
- If there are two terms, the polynomial will be called binomial
- If there are three terms, the polynomial will be called trinomial
The given trinomial is [tex]x^2 + 4x + 3[/tex]
Since the tiles are added to be in unit and positive(1, 2, 3, or 4), thus, let it be denoted by z, then:
[tex]x^2 + 4x + 3 + z = (x+a)^2[/tex]
since z is added such that the specified trinomial becomes perfect square trinomial, and since its degree is 2, so it must be product of two linear polynomials, and since coefficient of x in given trinomial is 1, so we kept the coefficient of linear polynomial as 1
Now, expanding the obtained equation, and comparing the coefficients, we get:
[tex]x^2 + 4x + 3 + z = (x+a)^2\\x^2 + 4x + 3 + z = x^2 + a^2 + 2ax\\[/tex]
Thus, from comparison of coefficients of like terms and constants, we get:
[tex]2a = 4\implies a = 2\\3 + z = a^2 = 2^2 = 4\\z = 1[/tex]
Thus, the number of unit tiles needed to be added to the expression [tex]x^2 + 4x + 3[/tex] for making it perfect square trinomial is: Option A: 1
Learn more about perfect square trinomials here:
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