8 (p – 3)^3+ 343 factorise it fast please
Answers
Answer:
Step-by-step explanation:
8(p-3)3-343
Final result :
(4p2 - 10p + 43) • (2p - 13)
Step by step solution :
Step 1 :
Equation at the end of step 1 :
8 • (p - 3)3 - 343
Step 2 :
2.1 Evaluate : (p-3)3 = p3-9p2+27p-27
Checking for a perfect cube :
2.2 8p3-72p2+216p-559 is not a perfect cube
Trying to factor by pulling out :
2.3 Factoring: 8p3-72p2+216p-559
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 8p3-559
Group 2: -72p2+216p
Pull out from each group separately :
Group 1: (8p3-559) • (1)
Group 2: (p-3) • (-72p)
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 :
2.4 Find roots (zeroes) of : F(p) = 8p3-72p2+216p-559
Polynomial Roots Calculator is a set of methods aimed at finding values of p for which F(p)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers p 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 8 and the Trailing Constant is -559.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4 ,8
of the Trailing Constant : 1 ,13 ,43 ,559
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -855.00
-1 2 -0.50 -686.00
-1 4 -0.25 -617.63
-1 8 -0.13 -587.14
-13 1 -13.00 -33111.00
-13 2 -6.50 -7202.00
-13 4 -3.25 -2296.13
-13 8 -1.63 -1134.45
-43 1 -43.00 -779031.00
-43 2 -21.50 -117992.00
-43 4 -10.75 -21139.88
-43 8 -5.38 -5042.42
-559 1 -559.00 -1420034967.00
-559 2 -279.50 -180362468.00
-559 4 -139.75 -23271519.38
-559 8 -69.88 -3096519.36
1 1 1.00 -407.00
1 2 0.50 -468.00
1 4 0.25 -509.38
1 8 0.13 -533.11
13 1 13.00 7657.00
13 2 6.50 0.00 2p-13
13 4 3.25 -342.88
13 8 1.63 -363.80
43 1 43.00 511657.00
43 2 21.50 50310.00
43 4 10.75 3380.88
43 8 5.38 -235.83
559 1 559.00 1375036585.00
559 2 279.50 169112034.00
559 4 139.75 20458072.38
559 8 69.88 2392319.11
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
8p3-72p2+216p-559
can be divided with 2p-13
Polynomial Long Division :
2.5 Polynomial Long Division
Dividing : 8p3-72p2+216p-559
("Dividend")
By : 2p-13 ("Divisor")
dividend 8p3 - 72p2 + 216p - 559
- divisor * 4p2 8p3 - 52p2
remainder - 20p2 + 216p - 559
- divisor * -10p1 - 20p2 + 130p
remainder 86p - 559
- divisor * 43p0 86p - 559
remainder 0
Quotient : 4p2-10p+43 Remainder: 0
Trying to factor by splitting the middle term
2.6 Factoring 4p2-10p+43
The first term is, 4p2 its coefficient is 4 .
The middle term is, -10p its coefficient is -10 .
The last term, "the constant", is +43
Step-1 : Multiply the coefficient of the first term by the constant 4 • 43 = 172
Step-2 : Find two factors of 172 whose sum equals the coefficient of the middle term, which is -10 .
-172 + -1 = -173
-86 + -2 = -88
-43 + -4 = -47
-4 + -43 = -47
-2 + -86 = -88
-1 + -172 = -173
1 + 172 = 173
2 + 86 = 88
4 + 43 = 47
43 + 4 = 47
86 + 2 = 88
172 + 1 = 173
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Final result :
(4p2 - 10p + 43) • (2p - 13)
Answer:
I have done it
Step-by-step explanation:
thank my answers