Respuesta :
For this case we have the following polynomial:
x2 + bx + c
To make the polynomial a perfect square, the value of c is:
c = (b / 2) ^ 2
Rewriting:
c = b ^ 2/2 ^ 2
c = b ^ 2/4
Substituting:
x2 + bx + b ^ 2/4
Where,
b is a constant.
Answer:
c = b ^ 2/4
x2 + bx + c
To make the polynomial a perfect square, the value of c is:
c = (b / 2) ^ 2
Rewriting:
c = b ^ 2/2 ^ 2
c = b ^ 2/4
Substituting:
x2 + bx + b ^ 2/4
Where,
b is a constant.
Answer:
c = b ^ 2/4
The value of 'c' which makes the given expression a perfect square trinomial is [tex]c =\dfrac{b^2}{4}[/tex]
What are perfect squares trinomials?
They are those expressions which are found by squaring binomial expressions.
How to find which of the given trinomials are perfect square trinomials?
Trinomials are polynomials with three terms or monomials.
Suppose the given trinomials are with degree 2, then, if they are perfect square, the binomial which was used to make them must be linear.
Let the binomial term was mx + d (a linear expression is always writable in this form where a and b are constants and m is a variable), then we will obtain:
[tex](mx+d)^2 = 2mxd + m^2x^2 + d^2[/tex]
What are like terms?
Those terms which have same variables raised with same power.
For example, [tex]5t^3[/tex] and [tex]6t^3[/tex] are like terms since variable is same, and it is raised to same power 3.
For example [tex]x^3[/tex] and [tex]3x^4[/tex] are not like terms as the variables are same but powers aren't same.
For the given case, the trinomial given is
[tex]x^2 + bx + c[/tex]
It has degree 2, and thus, if its a perfect square, it must be square of a binomial term. Let it be mx + b
Then, we get:
[tex](mx+d)^2 = 2mxd + m^2x^2 + d^2\\ x^2 + bx + c = m^2x^2 + 2mxd + d^2\\[/tex]
Comparing the coefficients of each like terms, and constants, we get:
For coefficients of [tex]x^2[/tex]:
[tex]m^2 = 1 \implies m=\pm 1[/tex]
For constant's comparison:
[tex]d^2 = c \implies d = \pm c[/tex]
Now, comparing the coefficients of [tex]x[/tex]:
[tex]2md = b\\2(\pm 1)(\pm \sqrt{c}) = b\\b = \pm 2\sqrt{c}\\\\c = \dfrac{b^2}{4}[/tex]
As when we compared the given expression with a perfect square trinomial, we assumed that c is a good fit. The value we obtained for c, thus, is a value for which the given expression is going to be a perfect square trinomial.
Thus, the value of 'c' which makes the given expression a perfect square trinomial is [tex]c =\dfrac{b^2}{4}[/tex]
Learn more about perfect square trinomials here:
https://brainly.com/question/2751022
