Muslim extremists are often the "mirror image" of the BNP, seeking out grievances to promote an "us and them" society, says David Cameron.
David Cameron has called for a calmer debate
To resist extremists of all sides, he said Britain must remove the "five barriers" dividing society.
These are uncontrolled immigration, extremism, multiculturalism, poverty and "educational apartheid", he says.
The Conservative leader also said in a speech people should be inspired, "not bullied", into feeling British.
His speech came as a poll for the centre-right Policy Exchange suggested support for Sharia law, Islamic schools and wearing the veil was much stronger among younger Muslims in Britain than among their parents.
Mr Cameron told the BBC Radio 4 Today programme: "It shows the extent to which multiculturalism has failed, because what the poll shows is that these young people feel more separated from Britain than their parents did."
In a speech to residents of Lozells, in Birmingham, later Mr Cameron will say that the ideology which inspired the 7 July bombers was one of the "great threats of our age".
He argues people must be drawn away from extremist ideologies - such as the British National Party on the one side and those Muslims who want to live under Sharia law on the other - which seek to point the blame at others and divide society, targeting the disaffected.
"Young white men are told: 'The blacks are all criminals. Young Afro-Caribbean men are told: 'The Asian shopkeepers are ripping you off'. Young Muslim men are told: 'The British want to destroy Islam'," Mr Cameron said in his speech.
"The best answer to ignorance like this is a good education."
He will say there was an "educational apartheid" between good schools and bad schools - some of the worst in deprived urban areas.
"A good education is important for everyone, but for children in poor areas it's absolutely vital," he said.
Poverty was also a major challenge - leading to other problems from family breakdown, unemployment, crime and addiction - with an "emerging underclass" of those left behind as the gap between rich and poor widens.
He said it is "an affront to social justice" as well as a "breeding ground for resentment and division".
Multiculturalism had reinforced divisions, treating communities as "monolithic blocks" not individuals and leading to social housing being allocated along ethnic lines, he said.
There is increasing translation of public documents into other languages, rather than incentives for people to learn English and "participate fully in our national life".
And ministers have to be in control of immigration, to stop it putting pressure on housing and services and creating more fear and resentment.
Mr Cameron also told the BBC ministers had been wrong to tell people to "plant flags in their lawns" - saying part of Britishness was "a sense of understatedness and restraint".
The government has launched a number of efforts to promote national identity and cohesion, including citizenship tests for foreigners who want to become British.
Chancellor Gordon Brown, widely expected to be Mr Cameron's opponent at the next general election, has also made "Britishness" one of his key themes.
Home Secretary John Reid says promoting better community relations helps tackle the rise of extremism, but created controversy last year when he urged British Muslims to "confront" extremist bullies in their communities.