Factorise: x^3 + 5x^2 +7x + 3
Answers
Answer:
Factoring: x3-5x2+7x-3
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 7x-3
Group 2: x3-5x2
Pull out from each group separately :
Group 1: (7x-3) • (1)
Group 2: (x-5) • (x2)
Bad news !! Factoring by pulling out fails :
The groups have no common factor and can not be added up to form a multiplication.
Polynomial Roots Calculator :
2.3 Find roots (zeroes) of : F(x) = x3-5x2+7x-3
Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers
The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient
In this case, the Leading Coefficient is 1 and the Trailing Constant is -3.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,3
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -16.00
-3 1 -3.00 -96.00
1 1 1.00 0.00 x-1
3 1 3.00 0.00 x-3
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms
In our case this means that
x3-5x2+7x-3
can be divided by 2 different polynomials,including by x-3
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : x3-5x2+7x-3
("Dividend")
By : x-3 ("Divisor")
dividend x3 - 5x2 + 7x - 3
- divisor * x2 x3 - 3x2
remainder - 2x2 + 7x - 3
- divisor * -2x1 - 2x2 + 6x
remainder x - 3
- divisor * x0 x - 3
remainder 0
Quotient : x2-2x+1 Remainder: 0
Trying to factor by splitting the middle term
2.5 Factoring x2-2x+1
The first term is, x2 its coefficient is 1 .
The middle term is, -2x its coefficient is -2 .
The last term, "the constant", is +1
Step-1 : Multiply the coefficient of the first term by the constant 1 • 1 = 1
Step-2 : Find two factors of 1 whose sum equals the coefficient of the middle term, which is -2 .
-1 + -1 = -2 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -1 and -1
x2 - 1x - 1x - 1
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-1)
Add up the last 2 terms, pulling out common factors :
1 • (x-1)
Step-5 : Add up the four terms of step 4 :
(x-1) • (x-1)
Which is the desired factorization
Multiplying Exponential Expressions:
2.6 Multiply (x-1) by (x-1)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (x-1) and the exponents are :
1 , as (x-1) is the same number as (x-1)1
and 1 , as (x-1) is the same number as (x-1)1
The product is therefore, (x-1)(1+1) = (x-1)2
Final result :
(x - 1)2 • (x - 3)
Answer:
Answer:
Here one root is x = 1 so (x - 1) is a factor. i.e
{x}^{3} - {x}^{2} - 3 {x}^{2} + 3x + 2x - 2 =x
3
−x
2
−3x
2
+3x+2x−2=
{x}^{2} (x - 1) - 3x(x - 1) + 2(x - 1) =x
2
(x−1)−3x(x−1)+2(x−1)=
(x - 1)( {x}^{2} - 3x + 2) =(x−1)(x
2
−3x+2)=
(x - 1)( {x}^{2} - 2x - x + 2) =(x−1)(x
2
−2x−x+2)=
(x - 1)(x(x - 2) - 1(x - 2)) =(x−1)(x(x−2)−1(x−2))=
(x - 1)(x - 2)(x - 1) = (x - 1) {}^{2} (x - 2)(x−1)(x−2)(x−1)=(x−1)
2
(x−2)