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It is required to decompose x^2+4x+3.
We shal treat the given expression of secon degree to consist of 2 factors like x+a and x+b. Then the product of the x+a and x+b must be the given expression. So.
(x+a)(x+b) = x^2+4x+3. So,
X^2+(a+b)x+ab = x^2+4x+3. Comparing the coefficients of x^2, x and constant terms on both sides, we get:
1/1 = 4/(a+b) = 3/(ab). Therefore,
a+b =4 and
Solving these inequality, you get, (We do it by guess here without prolonging the procedure): a = 3 and b =1.
Threfore, x^2+4x+3 = (x+3)(x+1)
I believe, by decomposed for of the expression you mean a form in which the expression is represented as multiple of two or more constituent factors.
The given expression is:
E = x^2 + 4x + 3
We can find its factors by suitably modifying the expression in stages without changing its value. For the given expression it can be done as follows:
E = x^2 + 4x + 3
= x^2 + 3x + x + 3 = (x^2 + 3x) + (x + 3)
= x*(x + 3) + 1*(x + 3)
= (x + 1)*(x + 3)
Thus factors of the given expression are (x + 1) and (x + 3).
We’ll consider the expression as a second grade polynomial.
A polynomial could be written as a product of linear factors, according to his roots. For this reason, first of all, we’ll try to find out the roots of the expression.
The coefficients of the polynomial x^2+4x+3=0 are a=1, b=4, c=3
x1=[-b + sqrt (b^2-4*a*c)]/2*a
x1=[-4 + sqrt (16-4*1*3)]/2*1
From Viete first relation between roots and coefficients, we’ll have:
-1 + x2=-4 => x2=-4+1
After finding the roots, we could write the expression as a product of linear factors.
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